The second edition of this book has a new title that more accurately reflects the table of contents. Over the past few years, many new results have been proven in the field of partial differential equations. This edition takes those new results into account, in particular the study of nonautonomous operators with unbounded coefficients, which has received great attention. Additionally, this edition is the first to use a unified approach to contain the new results in a singular place.Table of Contents: Markov semigroups in RN. Markov semigroups in unbounded open sets. A class of Markov semigroups in RN associated with degenerate elliptic operators. The nonautonomous setting. Appendices.