The first edition of thiook entitled Analysis on Riemannian Manifolds and Some Problems of Mathematical Physics was published by Voronezh University Press in 1989. For its English edition, the book haeen substantially revised and expanded. In particular, new material haeen added to Sections 19 and 20. I am grateful to Viktor L. Ginzburg for his hard work on the translation and for writing Appendix F, and to Tomasz Zastawniak for his numerous suggestions. My special thanks go to the referee for his valuable remarks on the theory of stochastic processes. Finally, I would like to acknowledge the support of the AMS fSU Aid Fund and the International Science Foundation (Grant NZB000), which made possible my work on some of the new results included in the English edition of the book.