A survey is given of a connection between compound matrices and ordinary differential equations. A typical result for linear systems is the following. If the n-th order differential equation
is uniformly stable, then a necessary and sufficient condition that the equation has an
-dimensional set of solutions satisfying
should be asymptotically stable. For nonlinear autonomous systems, a criterion for orbital asymptotic stability of a closed trajectory given by Poincaré in two dimensions is extended to systems of any finite dimension. A criterion of Bendixson for the nonexistence of periodic solutions of a two dimensional system is also extended to higher dimensions.