We provide a general framework for the understanding of Inexact Krylov subspace methods for the solution of symmetric and nonsymmetric linear systems of equations. In these methods, the matrix-vector product at each step is not performed exactly. This framework allows us to explain the empirical results reported in the literature, where the exactness of the matrix-vector product is allowed to deteriorate as the Krylov subspace method progresses. Furthermore, assuming exact arithmetic, our analysis produces computable criteria to bound the inexactness of the matrix-vector multiplication in such a way as to maintain the convergence of the Krylov subspace method. The theory developed is applied to several problems including the solution of Schur complement systems and linear systems which depend on a parameter. Numerical experiments are reported where the computable criteria are successfully applied.