Some algebraic geometry related to the mapping class group

This is a summary of the talk I gave in response to the request of the organizers of this workshop to survey the algebro-geometric aspects of the mapping class group. I am grateful to them for asking me to do so, as this led me to revisit (and to rethink) a few these connections. But the reader be warned that although I have tried to tailor this survey to an audience of geometers of a non-algebraic denomination, it remained one au goût du jour and consisted in the end more of asking questions than of stating results.