The recent development of Krylov subspace methods for solving linear systems of equations has shown that two fundamental approaches underlie the most commonly used algorithms: the Minimal Residual (MR) and Orthogonal Residual (OR) approaches. We show that these two approaches can be formulated as techniques for solving an approximation problem on a nested sequence of subspaces of a Hilbert space. Many well-known relations among the iterates and residuals of OR/MR pairs are shown to hold even in this abstract setting. When applied to the solution of linear systems of equations and when these subspaces are specialized to be a sequence of Krylov spaces, familiar OR/MR pairs result, among these CR/CG, GMRES/FOM and their specializations MINRES/CG to the Hermitian case, QMR/BCG as well as TFQMR/CGS. Further, we show that a common error analysis for these methods involving the canonical angles between subspaces allows many of the recently developed error bounds to be derived in a simple and uniform manner. An application of this analysis to operator equations involving compact perturbations of the identity shows that OR/MR pairs of Krylov subspace methods converge q-superlinearly for this class of problems.