Matrix equations of the one kind or the other are a central tool in all kinds of applications. In optimal control the linear quadratic control problem features a feedback solution that is determined via the solution of an algebraic Riccati equation. The system Gramians of a linear time invariant dynamical system are the solutions of two adjoint Lyapunov equations. Riccati, Lyapunov and Sylvester equations play an important role in different model order reduction techniques for continuous time linear dynamical systems. They all have discrete time counterparts as e.g. the well known Stein equations. Krylov subspace and eigenvalue methods can be related to certain Sylvester equations, as well. Nearly all other research field of the CSC group involve matrix equation computations at a certain point. The work of this team is dedicated to the efficient numerical solution of the variety of those equations in all kinds of working environments.