We present a unified theory of matrix pencil techniques for solving both continuous and discrete-time algebraic Riccati equations (ARE’s) under fairly general conditions on the coefficient matrices, The theory applies to a large class of ARE’s and Riccati-like equations arising from the singular H-infinity- and H-2- control problems, singular linear quadratic control, the ii-block Nehari problem, or from singular J-spectral factorizations. The underlying concept is the so-called proper deflating subspace of a (possibly singular) matrix pencil in terms of which necessary and sufficient conditions for the solvability of Riccati equations are given, It is shown that these conditions can be checked and the solutions computed by a numerically sound algorithm.