Second Order PDE’s in Finite and Infinite Dimension : A Probabilistic Approach The main objective of this monograph is the study of a class of stochastic differential systems having unbounded coefficients both in finite and in infinite dimension. We focus our attention on the regularity properties of the solutions and hence on the smoothing effect of the corresponding transition semigroups in the space of bounded and uniformly continuous functions. As an application of these results we study the associated Kolmogorov equations the large time behaviour of the solutions and some stochastic optimal control problems together with the corresponding Hamilton Jacobi Bellman equations. In the literature there exists a large number of works (mostly in finite dimen­ sion) dealing with these arguments in the case of bounded Lipschitz continuous coefficients and some of them concern the case of coefficients having linear growth. Few papers concern the case of non Lipschitz coefficients but they are mainly re­ lated to the study of the existence and the uniqueness of solutions for the stochastic system. Actually the study of any further properties of those systems such as their regularizing properties or their ergodicity seems not to be developed widely enough. With these notes we try to cover this gap. Kolmogorov equations in Rd with unbounded coefficients Asymptotic behaviour of solutions Analyticity of the semigroup in a degenerate case Smooth dependence on data for the SPDE: the Lipschitz case Kolmogorov equations in Hilbert spaces Smooth dependence on data for the SPDE: the non Lipschitz case (I) Smooth dependence on data for the SPDE: the non Lipschitz case (II) Ergodicity Hamilton Jacobi Bellman equations in Hilbert spaces Application to stochastic optimal control problems.