We construct Lévy processes with discontinuous jump characteristics in form of weak solutions of appropriate stochastic differential equations, or related martingale problems with non-local operators. For this purpose we prove a general existence theorem for martingale problems in which a sequence of operators generating Feller processes approximates an operator with a range containing discontinuous functions. The approach crucially depends on uniform estimates for the probability densities of the approximating processes derived from properties of the associated symbols. The theorem is applicable to stable like processes with discontinuous stability index. This talk is based on work with N. Willrich (WIAS Berlin).