Stochastic PDE’s and Kolmogorov Equations in Infinite Dimensions : Lectures given at the 2nd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Cetraro Italy August 24 September 1 1998 Kolmogorov equations are second order parabolic equations with a finite or an infinite number of variables. They are deeply connected with stochastic differential equations in finite or infinite dimensional spaces. They arise in many fields as Mathematical Physics Chemistry and Mathematical Finance. These equations can be studied both by probabilistic and by analytic methods using such tools as Gaussian measures Dirichlet Forms and stochastic calculus. The following courses have been delivered: N.V. Krylov presented Kolmogorov equations coming from finite dimensional equations giving existence uniqueness and regularity results. M. Röckner has presented an approach to Kolmogorov equations in infinite dimensions based on an LP analysis of the corresponding diffusion operators with respect to suitably chosen measures. J. Zabczyk started from classical results of L. Gross on the heat equation in infinite dimension and discussed some recent results