The aim of this paper is to study controllability of the linear infinite-dimensional system x = Ax + Bu, where A is the infinitesimal generator of a C 0-semigroup of linear bounded operators in the Banach space X; the control u is restricted to lie in a subset Ω of the Banach space U; Ω need not assumed to be convex or to contain 0 in its interior. Some necessary and sufficient conditions for approximate controllability to the whole space X are given. The proof of the main result is based on the spectral decomposition method developed by Fattorini and the generalized open mapping theorem due to Robinson. The obtained results are used to consider some controllability problems, with constrained controls, for the class of linear systems described by partial differential equations of parabolic type with bounded domain, and for the class of retarded functional differential equations in the state space M P . Particularly, in the case of the heat equation with positive scalar controls and in the case of the retarded equation with finite discrete delays, the general result leads to easily verifiable tests for approximate controllability, expressed in terms of the system matrices.