abstract: Modular forms have applications to several areas of mathematics e.g. elliptic curves, the theory of quadratic forms. Here we discuss a related class of functions, the so called weak Maass forms, which are related to Ramanujan's mock theta functions. In joint work with Ken Ono we prove that Dyson's rank generating functions, which include the mock theta function f(q) as a special case, are the ''holomorphic parts'' of weak Maass forms, the ''non-holomorphic parts'' are integrals of cuspidal weight 32 theta functions. As an application we obtain exact formulas for the coefficients of f(q), congruences for Dyson's ranks, asymptotics and inequalities for ranks, and identities for rank differences.