Presents a novel perspective of Markov semigroups, resulting in a better understanding of the relationships between Stochastic PDEs and Kolmogorov operators Special attention is paid to well-known models as the Ornstein-Uhlenbeck semigroup, the reaction-diffusion and Burgers equations SPDE. For each of them, the associated Kolmogorov operator is considered and the Kolmogorov equation for measures is solved. Moreover, a new characterization of the generator is given in the space if continuous functions by means of a core. Most of the results which can be found in the literature are obtained in functional spaces weighted by a suitable invariant measure for the transition semigroup. In this book no assumptions about ergodicity are given, and many properties of the transition semigroup and of the Kolmogorov operator are studied in spaces of continuous functions. The difficult problem of showing the uniqueness of a solution of the Kolmogorov equation for measure (also called Fokker-Planck or Kolmogorov backward) is solved.