Summary: We show that generalized approximation spaces can be used to describe the relatively compact sets of Banach spaces. This leads to compactness and convergence criteria in the approximation spaces themselves. If these spaces can be described with the help of moduli of smoothness, then the criteria can be formulated in terms of the moduli. As applications we give a generalization of Bernstein’s theorem about existence of elements with prescribed best approximation errors, compactness criteria for operators, a criterion for compactness in Sobolev type spaces, and a generalization of Simon’s compactness criterion for subsets of \(L^p\)-spaces of Banach-space-valued functions.