Nonlinear differential matrix equations generally stem from the semi-discretization on a rectangular grid of nonlinear partial differential equations (PDEs). The two main challenges related to approximating the solution of such matrix equations includes the high computational cost of time integrating the system when the matrices have large dimensions, as well as the cost related to evaluating the time-dependent nonlinear term at each timestep. In this presentation we give a brief overview of how model order reduction (MOR) techniques can be applied to lighten the computational load of approximating these matrix equations. Moreover, we consider in more detail the case where the nonlinear term is a special quadratic matrix function, better known as the differential Riccati equation (DRE). We show that great computational and memory advantages are obtained by order reduction methods onto fully rational Krylov subspaces and we discuss several crucial issues such as efficient.