We study the approximation of harmonic functions by universal overconvergent series. Most of the results established are analogues of those obtained in the case of approximation of holomorphic functions by such series. In the case of holomorphic functions, the approximation is made for functions which are continuous on a compact set and holomorphic inside this compact set, while our approximation is for functions that are harmonic in a neighborhood of the compact set. This difference is due to the fact that in the case of holomorphic functions, we have at our disposal Mergelyan's...