This book describes past and present results on Markov semigroups. It begins with the existence of solutions for elliptic and parabolic equations with unbounded coefficients on the whole of Rn . Then it continues with uniqueness and nonuniqueness results and regularity properties (e.g. compactness, uniform and pointwise estimates of the derivatives) of the associated semigroup, both on the space of bounded continuous functions and on Lp with invariant measure. One chapter is devoted to the Ornstein-Uhlenbeck operator as a prototype of an elliptic operator with unbounded coefficients. The second part of the book is devoted to elliptic and parabolic problems on open unbounded domains in Rn with Dirichlet and Neumann boundary conditions. The third part deals with degenerate problems. The monograph contains a very well-arranged collection of the results on Markov semigroups. It will be mainly appreciated by experts on Markov semigroups as well as researchers working in related topics. But not only by them, since the results are presented in a way suitable for applications. The book does not contain many examples but they are not necessary since the text is suitably understandable.