Introduct ion Let S be a closed subset of some open set in Cn and denote by dT(S) the space of germs of holomorphic functions on (a neighborhood of) S. For a space F(S) of tEvalued (continuous, differentiable etc.) functions on S [containing t~(S)] the problem of holomorphic approximation consists of finding conditions to ensure that the natural mapping Q : e)(S)-~F(S) has dense range with respect to a given topology on F(S). Positive solutions for F = C r, 0_ l . For Q:tP(/3)~O(D)c~C(/3), DCIE n strongly pseudoconvex, proofs were given independently by Henkin [17], Kerzman [21], and Lieb [27], for the case e : (9(/3)~(9(D)c~C~(/3) cf. also [30] and for Sobolev spaces see Bell [3, Sect. 6]. Moreover, supposing only parts of S to be totally real, H6rmander and Wermer 1-20] proved uniform holomorphic approximation of continuous functions that are holomorphic on the complement part, if S has a Stein neighborhood basis. For a detailed survey of known results we refer to Wells [39], Birtel [4], and Bedford and Fornaess [2]. The aim of this paper is to prove holomorphic approximation in spaces of ultradifferentiable functions [in the sense of Komatsu-Roumieu-Beurling, here a function f ~ C*(f2) is called ultradifferentiable if on compact subsets of O all partial derivatives of order p~lN can be estimated by a non-quasianalytic sequence Mp of positive real numbers satisfying some further conditions, cf. Sect. 1]. The spaces of ultradifferentiable functions are equipped with a natural locally convex topology which is finer than the C~-topology, so difficulties in holomorphic approximation of Mfdifferentiable functions arise from the fact that all derivatives have to be controlled simultaneously, in contrast to the Cr-case, 0 < r < 0o. We give a short summary of our results. After collecting the preliminaries [in particular the definition of ,-differentiability on submanifolds, here 9 stands either for (Mp) (Beurling case) or {Mp} (Roumieu case)] in Sect. 1, we prove in Sect. 2 that every