**Abstract**Many interesting systems in Physics and Applied Sciences are constituted by a large number of identical components so that they are difficult to be analyzed from a mathematical point of view. On the other hand, quite often, we are not interested in a detailed description of the system but rather to his collective behavior. Therefore it is necessary to look for all procedures leading to simplified models, retaining all the interesting features of the original system, cutting away redundant informations. This is exactly the methodology of the Statistical Mechanics and Kinetic Theory when dealing with large particle systems. Here we want to consider a particle system, as the basic starting point, to outline the limiting procedure leading to a more practical macroscopic description, usually expressed in term of a nonlinear partial differential equation. In this talk I discuss the most popular scalings, namely mean-field and low-density limits, leading to the Vlasov and Boltzmann equation respectively, starting from particle systems.

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