If X is a solution of a stochastic differential equation, in applications it is often of interest to calculate quantities of the form E(F(X(t))) for some chosen time t. A naive method for doing this is to use a Monte Carlo method where one simulates i.i.d. copies of X(t). But if one does not have the distribution of X(t), one needs to approximate X(t) with a numerical scheme, and then use a Monte Carlo method, thereby introducing another source of error. How to do this is fairly well understood for classic diffusions. If one replaces Brownian noise with more general semimartingale noise however, few results are known, and most are hard to use in practice. If, however, one replaces Brownian motion with a semimartingale which is also a Levy process, so that the solution X is still a strong Markov process, then one can obtain results similar to those of the Brownian case provided the jumps are restricted in magnitude. As one allows the jump sizes to increase, the rate of convergence of numerical methods slows down. We discuss recent results in this vein, obtained jointly with J.Jacod, T. Kurtz, and S. Meleard.