Faltings's theorem tells us that the set of rational points on a curve of genus at least 2 is finite, but explicitly determining this finite set is a difficult problem. In recent years, p-adic methods inspired by work of Chabauty, Coleman, and Kim have led to a flurry of activity on this front. In this talk, I will discuss some recent work in Chabauty's method for computing rational points. In particular, I will survey both geometric versions of the method and analytic methods for finding rational points using special values of p-adic L-functions.
