In this section we outline the derivation of the equation for the probability density function in the one-particle phase space for the smooth elastic hard spheres gas, that is the Boltzmann equation. In the kinetic theory [98] framework the derivation of transport equations begins from the Boltzmann equation [61]. In view of the interesting issues of Chapter 5 and of last section of Chapter 3, where we will address the problem of formation of ordered structures [214,213,215,212,178], that is (short and long range) spatial correlations, here we sketch the key passages for the derivation of the ring kinetic equation which relaxes the Molecular Chaos constraint and takes into account the recollision events, relevant in dense situations [4,83,82,74,75]. This section is devoted to the elastic gases, but the mainstream of the derivation is the same for the inelastic gases, and here we point out all the assumptions used which lack of rigorous proofs. At the end we discuss the Boltzmann-Enskog equation and the ring kinetic equation for inelastic smooth hard spheres [41,40,210,211], i.e. the main kinetic equations for granular gases.