This volume collects lecture notes from courses delivered in the past years at the Scuola Normale Superiore in Pisa, and also at the Trento and Funchal Universities. It presents an introductory course on differential stochastic equations and Malliavin calculus. The lectures are addressed to readers 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 the differential stochastic equations and their connection with parabolic problems. Several applications are given, notably the Feynman-Kac, Girsanov and Clark-Ocone formulae, the Krylov-Bogoliubov and Von-Neumann theorems. TOC:1. Gaussian measures in Hilbert spaces.- 2. L2 and Sobolev spaces w.r.t. a Gaussian measure.- 3. Brownian Motion.- 4. Markov property of the Brownian motion.- 5. The Itô integral.- 6. The Itô formula.- 7. Stochastic differential equations.- 8. Transition evolution operators.- 9. Formulae of Feynman-Kac and Girsanov.- 10. One dimensional Malliavin calculus.- 11. Malliavin calculus in more dimensions.- 12. Asymptotic behaviour of the transition semigroup.