This book is an attempt to give a systematic presentation of results and me- ods which concern the—xed point theory of multivalued mappings and some of its applications. In selecting the material we have restricted ourselves to stu- ing topological methods in the—xed point theory of multivalued mappings and applications, mainly to differential inclusions. Thus in Chapter III the approximation (on the graph) method in—xed point theory of multivalued mappings is presented. Chapter IV is devoted to the ho- logical methods and contains more general results, e.g. the Lefschetz Fixed Point Theorem, the—xed point index and the topological degree theory. In Chapter V applications to some special problems in—xed point theory are formulated. Then in the last chapter a direct applications to differential inclusions are presented. Note that Chapters I and II have an auxiliary character, and only results c- nected with the Banach Contraction Principle (see Chapter II) are strictly related to topological methods in the—xed point theory. In the last section of our book (see Section 75) we give a bibliographicalguide and also signalsome further results which are not contained in our monograph. The author thanks several colleagues and my wife Maria who read and c- mented on the manuscript. These include J. Andres, A. Buraczewski, G. Gabor, A. G´orka,M.Go´rniewicz, S. Park and A. Wieczorek. The author wish to express his gratitude to P. Konstanty for preparing the electronic version of this monograph.