  This survey is a preliminary version of a chapter of the forthcoming book
"Stochastic Analysis for Poisson Point Processes: Malliavin Calculus,
Wiener-It\^o Chaos Expansions and Stochastic Geometry" edited by Giovanni
Peccati and Matthias Reitzner. The paper develops some basic theory for the
stochastic analysis of Poisson process on a general $\sigma$-finite measure
space. After giving some fundamental definitions and properties (as the
multivariate Mecke equation) the paper presents the Fock space representation
of square-integrable functions of a Poisson process in terms of iterated
difference operators. This is followed by the introduction of multivariate
stochastic Wiener-It\^o integrals and the discussion of their basic properties.
The paper then proceeds with proving the chaos expansion of square-integrable
Poisson functionals, and defining and discussing Malliavin operators. Further
topics are products of Wiener-It\^o integrals and Mehler's formula for the
inverse of the Ornstein-Uhlenbeck generator based on a dynamic thinning
procedure. The survey concludes with covariance identities, the Poincar\'e
inequality and the FKG-inequality.
