We survey recent and also older results on nonsymmetric matrix Riccati differential equations and in the time invariant case on the corresponding algebraic Riccati equations. In particular we cite various applications connected with matrix Riccati equations. This paper reviews some basic results regarding the matrix Riccati equation of the optimal control and filtering theory. The theoretical exposition is divided into three parts dealing respectively with the steady-state algebraic equation, the differential equation, and the asymptotic properties of the solution. At the end a survey of existing computational techniques is given