This volume presents an introductory course on differential stochastic equations and Malliavin calculus. The material of the book has grown out of a series of courses delivered at the Scuola Normale Superiore di Pisa, as well as at Trento and Funchal Universities, and has been refined over several years of teaching experience in the subject. The lectures are addressed to readers who are familiar with basic notions of measure theory and functional analysis. The first part is devoted to the Gaussian measure in a separable Hilbert space, the Malliavin derivative, the construction of Brownian motion, and Itô's formula. The second part deals with differential stochastic equations and their connection with parabolic problems. The third part provides an introduction to Malliavin calculus. Several applications are given, notably the Feynman-Kac, Girsanov, and Clark-Ocone formulae.