This is the first of a five-volume exposition of the main principles of nonlinear functional analysis and its applications to the natural sciences, economics, and numerical analysis. The presentation is self-contained and accessible to the nonspecialist. Among the topics of Volume I are the two basic fixed-point theorems of Banach and Schauder, calculus in Banach spaces, the implicit function theorem, Newton's method, analytic bifurcation theory, fixed-point theorems for multivalued mappings, nonexpansive and condensing operators, mapping degree and fixed-point index and their applications, analytic maps, and asymptotic fixed-point theorems. The book contains numerous applications to such areas as ordinary and partial differential equations, integral equations, and game theory. Many exercises and a comprehensive bibliography complement the text.