diff --git a/docs/source/conf.py b/docs/source/conf.py
index 812a8ddf9..e6d7453d3 100644
--- a/docs/source/conf.py
+++ b/docs/source/conf.py
@@ -26,6 +26,10 @@
 bibtex_bibfiles = ["libsemigroups.bib"]
 
 autosummary_generate = True
+# We set this to True here, but remove "libsemigroups_pybind11\..*" from the doc
+# everywhere. This is done so we still get the submodule names, but not the
+# global module name. A nicer, but more involved solution, could use some Sphinx
+# magic as done in https://stackoverflow.com/a/72658470/15278419.
 add_module_names = True
 
 templates_path = ["_templates"]
diff --git a/docs/source/data-structures/misc/libsemigroups-error.rst b/docs/source/data-structures/misc/libsemigroups-error.rst
index 497eeebc8..d10c9c4e8 100644
--- a/docs/source/data-structures/misc/libsemigroups-error.rst
+++ b/docs/source/data-structures/misc/libsemigroups-error.rst
@@ -11,11 +11,15 @@ Exceptions
 ==========
 
 This page describes the custom error type used in ``libsemigroups_pybind11``,
-namely :any:`LibsemigroupsError`. Other built-in exceptions, such as
-:any:`ValueError` and :any:`TypeError`, may also be raised in this project. See
-the `Python documentation <https://docs.python.org/3/library/exceptions.html>`_
+namely :any:`LibsemigroupsError`, and related functions. Other built-in
+exceptions, such as :any:`ValueError` and :any:`TypeError`, may also be raised
+in this project. See the
+`Python documentation <https://docs.python.org/3/library/exceptions.html>`_
 for further details.
 
+Full API
+--------
+
 .. autoexception:: LibsemigroupsError
     :show-inheritance:
 
@@ -27,14 +31,14 @@ for further details.
 
         >>> from libsemigroups_pybind11 import FroidurePin, Perm
         >>> gens = [Perm([3, 0, 1, 2]), Perm([1, 2, 0, 3]), Perm([2, 1, 0, 3])]
-        >>> S = FroidurePin(gens[0])
-        >>> S.add_generators(gens[1:])
+        >>> S = FroidurePin(gens)
+        >>> S
         <partially enumerated FroidurePin with 3 generators, 3 elements, Cayley graph ⌀ 1, & 0 rules>
 
         >>> S.generator(3) # Bad: there are only three generators
         Traceback (most recent call last):
             ...
-        _libsemigroups_pybind11.LibsemigroupsError: generator index out of bounds, expected value in [0, 3), got 3
+        LibsemigroupsError: generator index out of bounds, expected value in [0, 3), got 3
 
     .. note::
 
@@ -43,3 +47,5 @@ for further details.
         but you think it should, please let us known by opening an issue on the
         `issue tracker
         <https://github.com/libsemigroups/libsemigroups_pybind11/issues>`_.
+
+.. autofunction:: error_message_with_prefix
\ No newline at end of file
diff --git a/docs/source/index.rst b/docs/source/index.rst
index effb8fb50..3f701a895 100644
--- a/docs/source/index.rst
+++ b/docs/source/index.rst
@@ -50,6 +50,63 @@ See the installation instructions:
     install
     changelog
 
+The structure of the module
+~~~~~~~~~~~~~~~~~~~~~~~~~~~
+The Python module ``libsemigroups_pybind11`` was designed to mirror the C++
+library ``libsemigroups`` as closely as possible, whilst navigating the
+fundamental difference between Python and C++. This is done with the help of the
+tool `pybind11 <https://pybind11.readthedocs.io/en/stable/>`_.
+
+For various implementational reasons (mostly related to Python's lack of an
+analogue for C++'s templating system), the Python code exposed by ``pybind11``
+is less streamlined and concise than the C++ library. To try and address this,
+the authors of ``libsemigroups_pybind11`` have further wrapped the Python code
+produced by ``pybind11`` to make the Python and C++ user experience as similar
+as possible.
+
+The Python bindings of the ``libsemigroups`` code produced by ``pybind11``
+reside in an intermediate module called ``_libsemigroups_pybind11`` (note the
+leading underscore), and the public-facing nicely wrapped code resides in this
+module — ``libsemigroups_pybind11``.
+
+Should this impact the way you, the user, use ``libsemigroups_pybind11``? For
+the most part, no. It should be possible to use ``libsemigroups_pybind11`` in
+the ways documented on this site without the knowledge that
+``_libsemigroups_pybind11`` even exists. The notable exceptions to this relate
+to type names and error messages. A lot of the code in
+``libsemigroups_pybind11`` has been imported from ``_libsemigroups_pybind11``,
+and this is visible in qualified type names. For example:
+
+.. doctest:: python
+
+    >>> from libsemigroups_pybind11 import WordGraph
+    >>> WordGraph
+    <class '_libsemigroups_pybind11.WordGraph'>
+
+Additionally, some functions or classes in the ``_libsemigroups_pybind11``
+module have additional prefixes and suffixes relative to their
+``libsemigroups_pybind11`` counterpart. These usually relate to the module that
+the function or class is contained within or a type the function or class is
+defined upon. These may appear in error messages. For example:
+
+.. doctest:: python
+
+    >>> from libsemigroups_pybind11 import aho_corasick, AhoCorasick
+    >>> ac = AhoCorasick()
+    >>> aho_corasick.add_word(ac, False)
+    Traceback (most recent call last):
+        ...
+    TypeError: aho_corasick_add_word(): incompatible function arguments. The following argument types are supported:
+        1. (ac: _libsemigroups_pybind11.AhoCorasick, w: list[int]) -> int
+        2. (ac: _libsemigroups_pybind11.AhoCorasick, w: str) -> int
+    <BLANKLINE>
+    Invoked with: <AhoCorasick with 1 node>, False
+
+The authors of ``libsemigroups_pybind11`` have gone to a lot of effort to
+try and make error messages clear, specific and intelligible; however, if there
+you encounter any errors with unclear messages, please raise this on the
+`issue tracker <https://github.com/libsemigroups/libsemigroups_pybind11/issues>`_.
+
 .. toctree::
     :caption: Data Structures
     :hidden:
diff --git a/etc/replace-strings-in-doc.py b/etc/replace-strings-in-doc.py
index fc4e8f498..df9f06650 100755
--- a/etc/replace-strings-in-doc.py
+++ b/etc/replace-strings-in-doc.py
@@ -18,8 +18,7 @@
 args = parser.parse_args()
 
 replacements = {
-    r"_libsemigroups_pybind11.": "",
-    r"libsemigroups_pybind11\.": "",
+    r'(?<=<span class="sig-prename descclassname"><span class="pre">)_?libsemigroups_pybind11\.': "",
 }
 
 html_glob = "docs/_build/html/**/*.html"
diff --git a/src/action.cpp b/src/action.cpp
index 6f93f9bab..db95199ff 100644
--- a/src/action.cpp
+++ b/src/action.cpp
@@ -147,7 +147,7 @@ Increase the capacity to a value that is greater or equal to *val*.
 :param val: new capacity of an action instance.
 :type val: int
 
-:raises ValueError:  if ``val`` is too large.
+:raises MemoryError:  if ``val`` is too large.
 
 :complexity:
   At most linear in the :any:`size()` of the :py:class:`Action`.
diff --git a/src/adapters.cpp b/src/adapters.cpp
index 6b52c22c2..8aac1ed15 100644
--- a/src/adapters.cpp
+++ b/src/adapters.cpp
@@ -102,7 +102,7 @@ This call operator returns the image of *pt* acted on by *x*.
 :rtype: Point
 
 :raises TypeError:
-    If the wrapped C++ type of the sample objects passed via *element* and
+    If the wrapped C++ type of the sample objects passed via *x* and
     *point* are not the same as the wrapped types of the arguments in any
     invocation of the call operator. For example, if *point* is ``PPerm([], [],
     256)``, then the underlying C++ type uses 8-bit integers to store image
@@ -171,7 +171,7 @@ This call operator returns the image of *pt* acted on by *x*.
 :rtype: Point
 
 :raises TypeError:
-    If the wrapped C++ type of the sample objects passed via *element* and
+    If the wrapped C++ type of the sample objects passed via *x* and
     *point* are not the same as the wrapped types of the arguments in any
     invocation of the call operator. For example, if *point* is ``PPerm([], [],
     256)``, then the underlying C++ type uses 8-bit integers to store image
diff --git a/src/dot.cpp b/src/dot.cpp
index 0b28374c7..3f61b73ba 100644
--- a/src/dot.cpp
+++ b/src/dot.cpp
@@ -52,10 +52,9 @@ render it with the `Graphviz <https://www.graphviz.org>`_
 installation on your system.
 )pbdoc");
     dot.def("__repr__", py::overload_cast<Dot const&>(&to_human_readable_repr));
-    dot.def_property(
+    dot.def_property_readonly(
         "colors",
         [](Dot const& self) { return Dot::colors; },
-        []() {},  // TODO raise exception
         R"pbdoc(
 An array of default HTML/hex colours.
 )pbdoc");
@@ -357,10 +356,9 @@ This nested class represents a node in the represented graph.
 )pbdoc");
     n.def("__repr__",
           py::overload_cast<Dot::Node const&>(&to_human_readable_repr));
-    n.def_property(
+    n.def_property_readonly(
         "attrs",
         [](Dot::Node& self) { return self.attrs; },
-        []() {},
         R"pbdoc(
      Read-only dictionary containing the attributes of the node.
      )pbdoc");
@@ -397,10 +395,9 @@ Instances of this nested class represents an edge in the represented graph.
 )pbdoc");
     e.def("__repr__",
           py::overload_cast<Dot::Edge const&>(&to_human_readable_repr));
-    e.def_property(
+    e.def_property_readonly(
         "attrs",
         [](Dot::Edge& self) { return self.attrs; },
-        []() {},
         R"pbdoc(
 Read-only dictionary containing containing the attributes of the :any:`Edge`.
      )pbdoc");
diff --git a/src/errors.cpp b/src/errors.cpp
index a66918ca1..4ad0df039 100644
--- a/src/errors.cpp
+++ b/src/errors.cpp
@@ -55,27 +55,55 @@ namespace libsemigroups {
   }
 
   void init_error(py::module& m) {
-    // TODO this doesn't seem to properly catch all LibsemigroupsExceptions,
-    // particularly on macOS. This may have been resolved in pybind11 2.12.0
-    static py::exception<LibsemigroupsException> exc(
-        m, "LibsemigroupsError", PyExc_RuntimeError);
+    // Using the GIL safe call below rather than simply having a static
+    // py::exception is recommended in the pybind11 doc.
+    PYBIND11_CONSTINIT static py::gil_safe_call_once_and_store<py::object>
+        exc_storage;
+    exc_storage.call_once_and_store_result([&]() {
+      return py::exception<LibsemigroupsException>(
+          m, "LibsemigroupsError", PyExc_RuntimeError);
+    });
     py::register_exception_translator([](std::exception_ptr p) {
       try {
         if (p) {
           std::rethrow_exception(p);
         }
       } catch (LibsemigroupsException const& e) {
-        exc(formatted_error_message(e).c_str());
-      } catch (py::stop_iteration const& e) {
-        throw e;
-      } catch (std::runtime_error const& e) {
-        exc(formatted_error_message(e).c_str());
+        py::set_error(exc_storage.get_stored(),
+                      formatted_error_message(e).c_str());
       }
     });
-    // TODO: Doc
+
     m.def("error_message_with_prefix",
-          py::overload_cast<>(&error_message_with_prefix));
+          py::overload_cast<>(&error_message_with_prefix),
+          R"pbdoc(
+Return whether :any:`LibsemigroupsError` messages have a C++ prefix.
+
+Since ``libsemigroups_pybind11`` is built on top of the C++ library
+``libsemigroups``, many of the errors thrown in ``libsemigroups_pybind11``
+emanate from C++. This function returns whether :any:`LibsemigroupsError`
+messages contain a prefix that indicates which C++ function raised the
+exception.
+
+:return: Whether :any:`LibsemigroupsError` messages contain a prefix about the
+  C++ function that raised the exception.
+:rtype: bool
+)pbdoc");
     m.def("error_message_with_prefix",
-          py::overload_cast<bool>(&error_message_with_prefix));
+          py::overload_cast<bool>(&error_message_with_prefix),
+          py::arg("val"),
+          R"pbdoc(
+Specify whether :any:`LibsemigroupsError` messages have a C++ prefix.
+
+Since ``libsemigroups_pybind11`` is built on top of the C++ library
+``libsemigroups``, many of the errors thrown in ``libsemigroups_pybind11``
+emanate from C++. This function specifies whether :any:`LibsemigroupsError`
+messages should contain a prefix that indicates which C++ function raised the
+exception. By default, this information is not included.
+
+:param val: Whether :any:`LibsemigroupsError` messages should contain a prefix
+  about the C++ function that raised the exception.
+:type val: bool
+)pbdoc");
   }
 }  // namespace libsemigroups
diff --git a/src/froidure-pin.cpp b/src/froidure-pin.cpp
index a822c3a60..fd8c318c7 100644
--- a/src/froidure-pin.cpp
+++ b/src/froidure-pin.cpp
@@ -132,6 +132,10 @@ Copy a :any:`FroidurePin` object.
 :returns: A copy.
 :rtype: FroidurePin
 )pbdoc");
+      // This function should really throw a ValueError if the degree of x is
+      // incompatible with the existing degree, but this doesn't get detected at
+      // the Python level, so a LibsemigroupsError is thrown instead. It would
+      // be possible to intercept this, but it probably isn't worth the effort.
       thing.def("add_generator",
                 &FroidurePin_::add_generator,
                 py::arg("x"),
@@ -166,7 +170,7 @@ elements than before (whether it is fully enumerating or not).
 :returns: ``self``.
 :rtype: FroidurePin
 
-:raises ValueError:
+:raises LibsemigroupsError:
    if the degree of *x* is incompatible with the existing degree (if any).
 
 :raises TypeError:
@@ -199,7 +203,7 @@ See :any:`add_generator` for a detailed description.
 :raises TypeError:
    if any item in *gens* is not of the same type as the existing generators (if any).
 
-:raises ValueError:
+:raises LibsemigroupsError:
    if the degree of any item in *gens* is incompatible with the existing degree (if any).
 )pbdoc");
 
diff --git a/src/libsemigroups_pybind11/action.py b/src/libsemigroups_pybind11/action.py
index c3947278d..26f3da1d5 100644
--- a/src/libsemigroups_pybind11/action.py
+++ b/src/libsemigroups_pybind11/action.py
@@ -209,7 +209,7 @@ def __init__(
         if _to_cxx(self) is not None:
             return
         if len(args) != 0:
-            raise ValueError(
+            raise TypeError(
                 f"expected 0 positional arguments, but found {len(args)}"
             )
         if not isinstance(generators, list):
diff --git a/src/libsemigroups_pybind11/adapters.py b/src/libsemigroups_pybind11/adapters.py
index 8df60bc7e..07673b4d9 100644
--- a/src/libsemigroups_pybind11/adapters.py
+++ b/src/libsemigroups_pybind11/adapters.py
@@ -68,9 +68,7 @@ def __init__(self: _Self, *args, point=None, element=None) -> None:
         if _to_cxx(self) is not None:
             return
         if len(args) != 0:
-            raise ValueError(
-                f"expected 0 positional arguments, but found {len(args)}"
-            )
+            raise TypeError(f"expected 0 positional arguments, but found {len(args)}")
         self.py_template_params = (
             type(_to_cxx(element)),
             type(_to_cxx(point)),
diff --git a/src/libsemigroups_pybind11/detail/cxx_wrapper.py b/src/libsemigroups_pybind11/detail/cxx_wrapper.py
index d44661ef1..b38a4e1a3 100644
--- a/src/libsemigroups_pybind11/detail/cxx_wrapper.py
+++ b/src/libsemigroups_pybind11/detail/cxx_wrapper.py
@@ -81,13 +81,13 @@ def __init__(
 
         for kwarg in required_kwargs:
             if kwarg not in kwargs:
-                raise ValueError(
+                raise TypeError(
                     f'required keyword argument "{kwarg}" not found, '
                     f"found {tuple(kwargs.keys())} instead"
                 )
         for kwarg in kwargs:
             if kwarg not in required_kwargs and kwarg not in optional_kwargs:
-                raise ValueError(
+                raise TypeError(
                     f'unexpected keyword argument "{kwarg}", '
                     f"required keyword arguments are {required_kwargs} "
                     f"and optional keyword arguments are {optional_kwargs}"
diff --git a/src/libsemigroups_pybind11/froidure_pin.py b/src/libsemigroups_pybind11/froidure_pin.py
index 69f401f37..2bb59a69f 100644
--- a/src/libsemigroups_pybind11/froidure_pin.py
+++ b/src/libsemigroups_pybind11/froidure_pin.py
@@ -162,7 +162,7 @@ def __init__(self: _Self, *args) -> None:
         if _to_cxx(self) is not None:
             return
         if len(args) == 0:
-            raise ValueError("expected at least 1 argument, found 0")
+            raise TypeError("expected at least 1 argument, found 0")
 
         if isinstance(args[0], list) and len(args) == 1:
             gens = args[0]
diff --git a/src/libsemigroups_pybind11/konieczny.py b/src/libsemigroups_pybind11/konieczny.py
index 51d527dea..860d69b70 100644
--- a/src/libsemigroups_pybind11/konieczny.py
+++ b/src/libsemigroups_pybind11/konieczny.py
@@ -130,7 +130,7 @@ def __init__(self: Self, *args) -> None:
         if _to_cxx(self) is not None:
             return
         if len(args) == 0:
-            raise ValueError("expected at least 1 argument, found 0")
+            raise TypeError("expected at least 1 argument, found 0")
 
         if isinstance(args[0], list) and len(args) == 1:
             gens = args[0]
diff --git a/src/libsemigroups_pybind11/presentation/__init__.py b/src/libsemigroups_pybind11/presentation/__init__.py
index b1634b1a1..6c72433b9 100644
--- a/src/libsemigroups_pybind11/presentation/__init__.py
+++ b/src/libsemigroups_pybind11/presentation/__init__.py
@@ -103,7 +103,7 @@ def __init__(self: Self, *args, **kwargs) -> None:
             or (len(args) == 1 and len(kwargs) > 0)
             or len(args) > 1
         ):
-            raise ValueError(
+            raise TypeError(
                 'expected 1 positional argument or the keyword argument "Word"'
                 f" but found {len(args)} positional arguments, and keywords arguments "
                 f"{tuple(kwargs.keys())}"
@@ -123,7 +123,7 @@ def __init__(self: Self, *args, **kwargs) -> None:
                     f"but found {type(args[0])}"
                 )
             if isinstance(args[0], list) and not all(isinstance(x, int) for x in args[0]):
-                raise ValueError("expected the argument to consist of int values")
+                raise TypeError("expected the argument to consist of int values")
             if isinstance(args[0], str):
                 self.py_template_params = (str,)
             if isinstance(args[0], list):
diff --git a/src/libsemigroups_pybind11/schreier_sims.py b/src/libsemigroups_pybind11/schreier_sims.py
index 6fa7b79b5..07cb943b1 100644
--- a/src/libsemigroups_pybind11/schreier_sims.py
+++ b/src/libsemigroups_pybind11/schreier_sims.py
@@ -71,7 +71,7 @@ def __init__(self: _Self, *args) -> None:
         if _to_cxx(self) is not None:
             return
         if len(args) == 0:
-            raise ValueError("expected at least 1 argument, found 0")
+            raise TypeError("expected at least 1 argument, found 0")
 
         if isinstance(args[0], list) and len(args) == 1:
             gens = args[0]
diff --git a/src/libsemigroups_pybind11/stephen.py b/src/libsemigroups_pybind11/stephen.py
index e6d9a2c3e..a97706079 100644
--- a/src/libsemigroups_pybind11/stephen.py
+++ b/src/libsemigroups_pybind11/stephen.py
@@ -73,7 +73,7 @@ def __init__(self: _Self, *args, **kwargs) -> None:
             return
 
         if len(args) != 1:
-            raise ValueError(f"expected 1 argument, but got {len(args)}")
+            raise TypeError(f"expected 1 argument, but got {len(args)}")
 
         if isinstance(args[0], (_Presentation, _InversePresentation)):
             self.py_template_params = (type(_to_cxx(args[0])),)
diff --git a/src/matrix.cpp b/src/matrix.cpp
index 9369855bd..fb887d0e2 100644
--- a/src/matrix.cpp
+++ b/src/matrix.cpp
@@ -27,7 +27,8 @@
 
 // libsemigroups....
 #include <libsemigroups/adapters.hpp>   // for Hash
-#include <libsemigroups/constants.hpp>  // for MaxPlusTruncMat, MinPlusTruncMat
+#include <libsemigroups/constants.hpp>  // for PositiveInfinity, NegativeInf...
+#include <libsemigroups/exception.hpp>  // for LIBSEMIGROUPS_EXCEPTION
 #include <libsemigroups/matrix.hpp>     // for MaxPlusTruncMat, MinPlusTruncMat
 
 #include <libsemigroups/detail/string.hpp>  // for string_format, to_string
@@ -408,7 +409,7 @@ Construct a matrix from rows.
 :param rows: the rows of the matrix.
 :type rows: list[list[int | PositiveInfinity | NegativeInfinity]]
 
-:raise RunTimeError: if *kind* is
+:raise TypeError: if *kind* is
     :py:attr:`MatrixKind.MaxPlusTrunc`,
     :py:attr:`MatrixKind.MinPlusTrunc`, or
     :py:attr:`MatrixKind.NTP`.
diff --git a/src/report.cpp b/src/report.cpp
index eab8e4192..adde609be 100644
--- a/src/report.cpp
+++ b/src/report.cpp
@@ -23,7 +23,7 @@
 #include <pybind11/pybind11.h>
 
 // libsemigroups_pybind11....
-#include "main.hpp"  // for init_error
+#include "main.hpp"  // for init_detail_report
 
 namespace libsemigroups {
 
diff --git a/src/word-range.cpp b/src/word-range.cpp
index 329c25443..7bbe7935a 100644
--- a/src/word-range.cpp
+++ b/src/word-range.cpp
@@ -26,6 +26,7 @@
 
 // libsemigroups....
 
+#include <libsemigroups/exception.hpp>   // for LIBSEMIGROUPS_EXCEPTION
 #include <libsemigroups/paths.hpp>       // for Path
 #include <libsemigroups/types.hpp>       // for word_type
 #include <libsemigroups/word-range.hpp>  // for number_of_words
@@ -1423,7 +1424,7 @@ documentation of that function for more details.
 :returns: The human readable index.
 :rtype: int
 
-:raises LibsemigroupsError: if *c* is not exactly one character long.
+:raises ValueError: if *c* is not exactly one character long.
 
 .. seealso::
     :any:`words.human_readable_letter`
diff --git a/tests/test_schreier_sims.py b/tests/test_schreier_sims.py
index 57fcfde1c..39b4ca02d 100644
--- a/tests/test_schreier_sims.py
+++ b/tests/test_schreier_sims.py
@@ -29,7 +29,7 @@
 def check_constructors(gens):
     ReportGuard(False)
     # default constructor
-    with pytest.raises(ValueError):
+    with pytest.raises(TypeError):
         SchreierSims()
 
     S1 = SchreierSims(gens)
diff --git a/tests/test_words.py b/tests/test_words.py
index 6acfda46c..440cbe0ab 100644
--- a/tests/test_words.py
+++ b/tests/test_words.py
@@ -348,5 +348,5 @@ def test_human_readable_letter():
 
 def test_human_readable_index():
     assert human_readable_index("\377") == 255
-    with pytest.raises(LibsemigroupsError):
+    with pytest.raises(ValueError):
         human_readable_index("Ā")  # this is character 256