In this paper we obtain a Carleman approximation theorem for maps from Stein manifolds to Oka manifolds. More precisely, we show that under suitable complex analytic conditions on a totally real set M of a Stein manifold X, every smooth map X → Y to an Oka manifold Y satisfying the Cauchy-Riemann equations along M up to order k can be C k -Carleman approximated by holomorphic maps X → Y . Moreover, if K is a compact O(X)-convex set such that K ∪ M is O(X)-convex, then we can C k -Carleman approximate maps which satisfy the Cauchy-Riemann equations up to order k along M and are holomorphic on a neighbourhood of K, or merely in the interior of K if the latter set is the closure of a strongly pseudoconvex domain.

Let R be an open Riemann surface. In this paper we prove that every continuous function M → R n , n ≥ 3, defined on a divergent Jordan arc M ⊂ R can be approximated in the Carleman sense by conformal minimal immersions; thus providing a new generalization of Carleman's theorem. In fact, we prove that this result remains true for null curves and many other classes of directed holomorphic immersions for which the directing variety satisfies a certain flexibility property. Furthermore, the constructed immersions may be chosen to be complete or proper under natural assumptions on the variety and the continuous map.As a consequence we give an approximate solution to a Plateau problem for divergent Jordan curves in the Euclidean spaces.