The finite volume method is a discretization method for solving partial differential equations (PDE) where the degrees of freedom approximate the average of the PDE solution over control volumes. In this talk, we will apply this method to the Euler equations, a system of non-linear hyperbolic PDEs governing the dynamics of a compressible, adiabatic and inviscid fluid. Particular attention will be paid to the robustness and stability of the approximation and to ensure, at the discrete level, some fundamental physical principles (e.g., conservation, positivity of some quantities, second law of thermodynamics).