The geometry of the moduli space of curves and abelian varieties

Paper in proceeding, 2017

Both algebraic curves and abelian varieties are basic objects in algebraic geometry and the study of their moduli spaces is a classical topic, with connections to number theory and other areas of geometry. In these expository notes, we discuss and compare the main geometric properties of moduli spaces of curves and abelian varieties, also in view of the relationships between them arising from Torelli and Prym-type maps.