[For the entire collection see Zbl 0746.00063.]
The aim is to establish Malliavin calculus directly on any probability space endowed with a space of Gaussian variables, rather than on an abstract Wiener space. This calculus is based on a set of simple variables which are polynomial functions of the Gaussian variables; gradient, divergence and Ornstein-Uhlenbeck operators are defined and studied in this setting. Groups of operators associated to the Ornstein- Uhlenbeck operator are used to define Sobolev norms and generalized functions. This paper is written with precise definitions and proofs, so that it is a self-contained exposition of the subject.