Abelian varieties are compact complex tori that can be embedded in projective space. They have been studied in depth since the 19th century, going back to the work of Riemann, Abel, Jacobi and others. Among the most well-understood examples are Jacobians of algebraic curves. Prym varieties are abelian varieties associated to connected, unramified covers of algebraic curves. While a great deal is known about them, there remain a number of interesting open questions. In this talk I will discuss the background on Prym varieties, including some of the applications in algebraic geometry that drew people's attention to these examples in the 1970s, and will also describe some open problems. I will then discuss some recent progress, including joint work with Samuel Grushevsky, Klaus Hulek, and Radu Laza on extending the Prym map to toroidal compactifications of the moduli of abelian varieties.