Abelian varieties are those algebraic varieties which are both compact and have a group structure. They are interesting in their own right, but can also be used to study other algebraic varieties through the construction of the Picard or Albanese varieties. After giving some basics on abelian varieties, we specialize to the "principally polarized" case, i.e., those abelian varieties with a theta divisor. Then we will see results concerning the singularities of these divisors and explain how this helps to understand which abelian varieties could possibly arise as Picard varieties of curves. Speaker(s): Bradley Dirks (UM)