This work was originally published in 1973, and appeared in a second edition in 1982 (Birkhauser), of which this is a reprint. Covering the basics of the theory of ordinary differential equations, discussion centers on the integration of differential inequalities but also describes techniques involving simple topological arguments, fixed point theorems, and functional analysis. The text assumes a familiarity with matrix theory and the ability to deal with functions of real variables. Hartman taught mathematics at Johns Hopkins University.