The derivation of effective descriptions from microscopic dynamics is a very vivid area in mathematical physics. In the talk I will discuss a system of many particles with Newtonian time evolution that are subject to interaction. It is well known that in the weak coupling limit this system converges, under smoothness assumption on the interaction force, to a solution of the Vlasov equation. Weakening the types of convergence (convergence for all initial conditions → convergence in probability → convergence in distribution), the smoothness condition on the interaction can be generalized. In the talk I will present recent results in this direction and explain which types of convergence hold or do not hold under the different assumptions on the interaction force.