In this chapter, we consider large-scale continuous-time differential Riccati and Lyapunov matrix equations which are ubiquitous in optimal control and model order reduction problems. Only a few methods have been proposed to numerically solve these equations, unlike their well-studied algebraic counterparts. This work is based on a series of papers recently published by the authors. Our approach is based on the reduction of the problem dimension prior to integration. Projecting the initial problem onto a sequence of nested block Krylov subspaces, we obtain, at each step, a low-dimensional differential matrix equation, which is solved by classical numerical integration schemes. When the expected accuracy is reached, an approximate solution of the original problem is reconstructed, taking advantage of the low-rank structure of the problem. We give some theoretical results and, when possible, a way to contain the dimension of the projection space and reduce the computational time without affecting the accuracy.