For an open subset U of a locally convex space E, let (H(U), tau(0)) denote the vector space of all holomorphic functions on U, with the compact-open topology. If E is a separable Frechet space with the bounded approximation property, or if E is a (DFC)-space with the approximation property, we show that (H(U), tau(0)) has the approximation property for every open subset U of E. These theorems extend classical results of Aron and Schottenloher. As applications of these approximation theorems we characterize the spectra of certain topological algebras of holomorphic mappings with values in a Banach algebra. (C) 2003 Elsevier Inc. All rights reserved.1262141156