In this seminar it will be presented the construction of the macroscopic evolution equations of a reacting gas mixture starting from the microscopic description of the particles interaction. Such a construction is based on the reactive Boltzmann equation that describes the evolution of the distribution function defined in the phase space, which indicates the position and velocity of each particle at a given instant. Then, this macroscopic framework is used to investigate the dynamics of the steady detonation wave and study the linear hydrodynamic stability of the steady detonation solutions. With reference to a binary reactive mixture, whose particles may undergo binary elastic scattering and reactive collisions according to the chemical law A+A?B+B, the results presented in this seminar include the modeling of the steady detonation waves, its hydrodynamic stability and the numerical treatment of these problems.