Covers the fundamentals of the theory of ordinary differential equations (ODEs), including an extensive discussion of the integration of differential inequalities, on which this theory relies heavily. In addition to these results, the text illustrates techniques involving simple topological arguments, fixed point theorems, and basic facts of functional analysis. Unlike many texts, which supply only the standard simplified theorems, Ordinary Differential Equations presents the basic theory of ODEs in a general way, making it a valuable reference. This SIAM reissue of the 1982 second edition covers invariant manifolds, perturbations, and dichotomies, making the text relevant to current studies of geometrical theory of differential equations and dynamical systems. In particular, Ordinary Differential Equations includes the proof of the Hartman-Grobman theorem on the equivalence of a nonlinear to a linear flow in the neighborhood of a hyperbolic stationary point, as well as theorems on smooth equivalences, the smoothness of invariant manifolds, and the reduction of problems on ODEs to those on maps (Poincare). Ordinary Differential Equations is based on the author's lecture notes from courses on ODEs taught to advanced undergraduate and graduate students in mathematics, physics, and engineering. The book, which remains as useful today as when it was first published, includes an excellent selection of exercises varying in difficulty from routine examples to more challenging problems. These exercises show extensions of the techniques in question and serve to introduce the reader to the literature in this area. New to this SIAM Classics edition is an Errata section listing corrections to minor errors in the 1982 edition. An extensive bibliography (up to 1980) is provided.