Systems of interacting particles describing notable physical phenomena, such as Bose-Einstein condensation, superconductivity or superfluidity, exhibit a daunting complexity. This complexity renders the exact many-body theory computationally non-approachable, even for physicists conducting computer experiments and simulations. Therefore, an approximate description using effective macroscopic models is highly useful, and the rigorous study of the regime of validity of such approximations is of primary importance in mathematical physics. In this talk we will focus on the dynamics of systems made of quantum particles in the mean-field regime, where weakly interacting particles exhibit a collective behavior approximated by an averaged potential in convolution form. We study time scales where the semiclassical description becomes relevant. Through a novel technique based on weak-strong stability principles for partial differential equations, we show that the many-body dynamics is well approximated by a Vlasov equation describing the evolution of the effective probability density of particles on the phase space.