The objective of these lectures is to present Ornstein-Uhlenbeck and related stochastic processes to a wide mathematical audience with a modest preparation in stochastic analysis. The aim of the first part of the lectures (Chapter 1) is to discuss the Ornstein-Uhlenbeck and the Squared Radial Ornstein-Uhlenbeck stochastic diffusion processes, whose infinitesimal generators are, respectively, the Ornstein-Uhlenbeck operator and the Laguerre operator. In the second chapter of these lectures the Ornstein-Uhlenbeck processes governed by -stable rotationally invariant processes are studied. This corresponds to replacing the Laplacian by the fractionnary Laplacian -(-)^/2 in the Ornstein-Uhlenbeck generator L=12-x. More general drift terms b(x) are also considered at the end of Chapter 2.[2mm]