This volume presents an introductory course on differential stochastic equations and Malliavin calculus. The material of the book has grown from a series of courses delivered at the Scuola Normale Superiore of Pisa (and also at the Universities of Trento and Funchal) and has been refined over several years of teaching experience in the subject. The lectures are addressed to a reader who is 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 the Brownian motion and Itô’s formula. The second part deals with differential stochastic equations and their connection with parabolic problems. The third part contains an introduction to Malliavin calculus. Several applications are given, notably the Feynman-Kac, Girsanov and Clark-Ocone formulae, the Krylov-Bogoliubov and Von Neumann theorems.