This paper is a brief survey of the recent results in problems of approximating functions by solutions of homogeneous elliptic systems of PDEs on compact sets in the plane in the norms of C^m spaces, m ≥ 0. We focus on general second-order systems. For such systems the paper complements the recent survey by M. Mazalov, P. Paramonov, and K. Fedorovskiy (2012), where the problems of C^m approximation of functions by holomorphic, harmonic, and polyanalytic functions as well as by solutions of homogeneous elliptic PDEs with constant complex coefficients were considered.