Abstract: I will talk about numerical simulations of structure formation using fast approximate methods in both LCDM and alternative theories. N-body simulations of modified gravity have been with us for a decade or so. We know how to do them, but they are still generally much slower than a corresponding LCDM simulation. I will present a simple approximate method for general models that has a screening mechanism that allows for much faster N-body simulations. Then I will move on to some approximate methods of traditional N-body itself. N-body is too computationally expensive for many purposes, for example if one wants to generate large ensembles of galaxy mocks to model the observables and their covariances as needed for future surveys. This is where approximate methods comes in handy. I will show how one can extend the fast COLA method to models where the growth-rate is scale-dependent like it is in most modified gravity theories. Another application of this is that it allows for a simple way to include the effects of massive neutrinos in COLA.