After introducing the Brouwer degree theory in Rn, the authors consider the Leray-Schauder degree for compact mappings in normed spaces. A description of degree theory for condensing mappings (chapter 3) is followed by a chapter dedicated to studies of degree theory for A-proper mappings. The focus then turns to the construction of the Mawhin coincidence degree for L-compact mappings, degree theory for mappings of class (S+) and their perturbation with other monotone-type mappings. The last chapter is dedicated to fixed point index theory in a cone of a Banach space and presents a new fixed point index for countably condensing maps. Each chapter is accompanied by important applications illustrating the reason that it was necessary to change the previous concept of topological degree to a more general one. Many examples and exercises conclude each chapter. The book forms a good text for a self-study course or special topic courses and it is an important reference for anybody working in differential equations, analysis or topology. In summary, we can say that the book is an up-to-date exposition of the theory and applications of an important part of mathematics.