Hailed by The American Mathematical Monthly as "a rigorous and lively introduction," this text explores a topic of perennial interest in mathematics. The author, a distinguished mathematician and formulator of the Hurewicz theorem, presents a clear and lucid treatment that emphasizes geometric methods. Topics include first-order scalar and vector equations, basic properties of linear vector equations, and two-dimensional nonlinear autonomous systems. Suitable for senior mathematics students, the text begins with an examination of differential equations of the first order in one unknown function. Subsequent chapters address systems of differential equations, linear systems of differential equations, singularities of an autonomous system, and solutions of an autonomous system in the large.