The derivation of effective macroscopic theories approximating microscopic systems of interacting particles in some scaling limit is a major question in non-equilibrium statistical mechanics. In this talk we will be concerned with the dynamics of systems made of many interacting fermions and focus on the mean-field regime, i.e. weakly interacting particles whose collective effect can be approximated by an averaged potential in convolution form. As a first step we will obtain a reduced description given by the time-dependent Hartree-Fock equation. As a second step we will look at longer time scales, where a semiclassical description starts to be relevant, and approximate the many-body dynamics with the Vlasov equation, which describes the evolution of the effective probability density of particles on the one particle phase space. The structure of the initial data will play an important role at each step of the approximation.