Malliavin Calculus, an infinite dimensional calculus on probability spaces, was born with the paper

P. Malliavin - Stochastic calculus of variations and hypoelliptic operators, Proc. Inter. Symp. Stoch. Diff. Eqs. Kyoto 1976, Wiley (1978), 195–263.

and was initially aimed to give a probabilistic counterpart of Hörmander theorem for hypoelliptic operators. Soon it found many developments and other applications within Mathematics (notably in Mathematical Physics), also in Finance.

I will give an introduction to Malliavin Calculus techniques, with a brief reference to some applications.