For discretizations of partial differential equations, the standard methods for numerically solving Lyapunov and Riccati equations (as for example implemented in matlab) are usually not suitable since the computational effort and storage requirements are too high. Analysis of the underlying partial differential equation shows that often good low rank approximations to the exact solution of the Lyapunov or Riccati equation exist. Such low rank approximations are cheap to store and can often be efficiently computed. This talk will describe some recent work on approximately solving Lyapunov and Riccati equations for (discretizations of) partial differential equations.