We give the complete solution of a large class of problems in linear system theory, the so-called cover problems. These problems are formulated and solved both in the state-space and in the input-output frameworks. The key concept, which allows the effective parametrization of all solutions of the cover problems, is that of the partial realizations of a sequence of matrices. It is shown that the solutions of the state-space cover problems can be expressed as state spaces of the partial realizations of appropriately defined sequences of matrices, and the solution of the input-output cover problems can be expressed as a simple function of the functions of the partial realizations of the sequences of mentioned above.