Abstract: Let C be a Riemann surface, x_1,...,x_n a set of points on C, and a_1,...,a_n integers adding up to 0. The theorem of Abel and Jacobi determines when C carries a meromorphic function with zeroes and poles at the points x_i,  with multiplicities given by a_i. Modern versions of the theorem ask the same question in families -- to determine the locus of curves that admit meromorphic functions with prescribed zeroes and poles inside the moduli space of curves. In this talk, I will explain how recent progress in the intersection theory of normal crossings pairs leads to a solution of this and several related problems.