The other answers are great. But I thought I (the OP) ought to share what I do these days (a year or two after the question).

I use Sphinx and its Markdown extension. Do the following:

TL;DR: See Gist snippet.

Sphinx-markdown-builder

You need sphinx-markdown-builder python module.

 pip install sphinx sphinx-markdown-builder;

Run Sphinx

Not the autodoc, the apidoc!

sphinx-apidoc -o Sphinx-docs . sphinx-apidoc --full -A 'Matteo Ferla'; cd Sphinx-docs;

Configuration

Fix the conf.py file, by following the following or just lazily copy paste the echo command below.

Manual

First uncomment the lines. These are otherwise commented out.

import os
import sys
sys.path.insert(0, os.path.abspath('../'))

Note the change to ../

One weirdness is that the magic methods get ignored. To override this, add this anywhere:

def skip(app, what, name, obj, would_skip, options):
    if name in ( '__init__',):
        return False
    return would_skip
def setup(app):
    app.connect('autodoc-skip-member', skip)

A thing to note: The docstrings ought to be written in restructuredtext (RST). If they are in Markdown, you need to add a mod – see this. The two are similar, but different. For example, a single backquote is required for <code> in Markdown, while two are for RST. If in doubt, several blog posts discuss the merits of RST documentation over Markdown.

Typehinting

RST typehints (:type variable: List) are obsolete as proper typehinting def foo(variable: Optional[List[int]]=None) -> Dict[str,int]: has been introduced since 3.6. To make these work:

 pip install sphinx-autodoc-typehints

And add 'sphinx_autodoc_typehints' at the end of the extensions list. Note the package has hyphens while the module has underscores.

TL;DR

Copy paste this:

echo " import os
import sys
sys.path.insert(0,os.path.abspath('../'))
def skip(app, what, name, obj,would_skip, options):
    if name in ( '__init__',):
        return False
    return would_skip
def setup(app):
    app.connect('autodoc-skip-member', skip)
extensions.append('sphinx_autodoc_typehints')
 " >> conf.py;

Showtime

Then it is showtime.

make markdown;

Copy the files and clean however you fancy.

mv _build/markdown/* ../; rm -r Sphinx-docs;

Repeat Apidoc for new files

It should be noted that when new files are added, the apidoc command needs to be repeated. Nevertheless, I highly recommend generating documentation midway as I often realise I am doing something wrong when I see the docs.

But briefly, apidoc will add for each file a automodule command, so this could be added manually or even expanded:

.. automodule:: my_module
   :members:
   :inherited-members:
   :undoc-members:
   :show-inheritance:

There’s also the commands autoclass, autofunction, autoexception, for specific cases. In the case of autoclass if the class inherits many base classes in separate files to rightfully keep filesizes under 250 lines, the property :inherited-members: is a nice addition to this —thus avoiding having to describe the private base classes.

Read the docs: the common way

It should be said that there’s a trend to not have documentation in GitHub but in Read the docs. My guess is because:

• avoids this docstrings-to-markdown business
• some users get confused by GitHub
• looks nicer
• other do it

Despite this, it requires some set up due to the module requirements. In another SO post is a long list of pitfalls and tricks —briefly IMO users, such as myself, make three mistakes:

1. missing modules or the target module
2. forget to hard refresh the browser
3. enabling the sphinx.ext.autodoc extension

However, if one has written markdown documentation in GitHub these can be imported too. Formerly, the m2r2 (a fix of the deprecated m2r) was a good solution, but the divergence of its dependency mistune, which would require it to be frozen at version 0.8.4 as opposed to being at 2.0.0, which breaks other sphinx modules, therefore a new split works best and better: sphinx-mdinclude.
This is pip installed as sphinx-mdinclude but included as sphinx_mdinclude and allows md files to be read alongside rst files.
So a simple workaround in the docs/source/config.py file is to copy the files from the project root to the folder of config.py One issue is that links may need to be checked, especially if files moved around or are relative to the base URL (slash prefixed), eg. [Foo](/md_docs/foo).