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Since the 1960s, many researchers have extended topological degree theory to various non-compact type nonlinear mappings, Presenting a survey of advances made in generalizations of degree theory during the past decade, this book focuses on topological degree theory in normed spaces and its applications. It includes discussions on a degree theory for new monotone maps and A-proper Fredholm maps of index-zero type and presents a recently developed fixed point index for countably condensing maps. Applications to ordinary and partial differential equations and evolution equations are presented throughout the book, and each chapter includes exercises suitable for self-study and special topics courses.