Algebraic curves and abelian varieties play a central role in modern algebraic geometry, with links to complex analysis, number theory, topology and others. Curves and abelian varieties are closely related: a fundamental example of an abelian variety is the Jacobian of an algebraic curve. In this talk, I will give a discussion of curves, abelian varieties and their moduli spaces. Time permitting, I will present some new tools aimed at studying geometric classes on the moduli space of abelian varieties, and conclude with a discussion of several open questions in this area.