Curves and their Jacobians: can we find algebraic curves (say over the complex numbers) such that their Jacobians have complex multiplication? Can we relate natural structures on M_g to natural structures on A_g ? We study these questions, and we soon find out that there are more open problems than theorems. -- We will discuss some recent progress, mainly due to Ben Moonen, and we pose many intriguing open questions. See my notes for this talk, which is mainly taken from three papers:

F. Oort -- Moduli of CM abelian varieties.

B. Moonen & F. Oort -- The Torelli locus and special subvarieties.

B. Moonen - - Special subvarieties arising from families of cyclic covers of the projective line.