I will consider supersymmetric field theories on compact manifolds M and obtain constraints on the dependence of their partition functions Z_M on the geometry of M. For N=1 theories with a U(1) R symmetry in four dimensions, M must be a complex manifold with a Hermitian metric. I will show how to describe the theory in terms of twisted variables that make easy to analyze the dependence of Z_M on the parameters entering the Lagrangian. I will also show that Z_M is "almost" topological: Z_M is independent of the Hermitian metric and depends holomorphically on the complex structure moduli.