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.