Dark matter in the Universe can be considered as a collisionless self-gravitating fluid obeying the Vlasov-Poisson equations. In the standard picture of cosmic structure formation, the first dark matter objects to form are expected to be microhalos of roughly Earth mass and solar system size. These halos can subsequently merge to form larger dark matter halos such as that of our Galaxy. In practice, resolving dark matter dynamics relies on a N-body approach, but with the advent of exaflopic computers it now becomes possible to solve directly Vlasov dynamics in six-dimensional phase-space. In this talk, I will introduce the general concepts about these numerical methods through a visual comparison between particle-mesh simulations to Vlasov runs in the framework of the formation of dark matter microhalos. I will subsequently discuss recent results obtained for the profile of microhalos, notably the appearance of "prompt cusps" which could survive until the present time and increase the probability of indirect detection of dark matter compared to standard calculations. I will also describe how the very early stages of the microhalo dynamics can be addressed accurately with high order perturbation theory and how this could be linked to establishing a self-similar pattern via violent relaxation.