Topological Amplitudes and Physical Couplings in String Theory

In these lectures, we review the main properties of the topological theory obtained by twisting the N=2 two-dimensional superconformal algebra, associated to supersymmetric string compactifications. In particular, we describe a set of physical quantities in string theory that are computed by topological amplitudes. These are in general higher-dimensional F-terms in the low energy effective supergravity, or fermion masses after supersymmetry breaking. We discuss N=2 compactifications of type II strings, N=1 compactifications of heterotic and type I strings, as well as N=4 string vacua. Particular emphasis is put on alternative string dual representations allowing calculability, and on the generalization of N=2 holomorphicity and its anomaly.