The Euler equation of gas dynamics are frequently used in engineering, meteorology and fluid dynamics. Recent theoretical results imply that the Euler equations are ill-posed in the class of weak solutions satisfying entropy inequality. Consequently, the question of convergence of numerical methods is fundamental. I will present our recent results on the convergence analysis of suitable finite volume methods for multidimensional Euler equations. We have shown that a sequence of numerical solutions converges weakly to a weak dissipative solution. The analysis requires only the consistency and stability of a numerical method and can be seen as a generalization of the famous Lax-equivalence theorem for nonlinear problems. The weak-strong uniqueness principle implies the strong convergence of numerical solutions to the classical solution as long as it exists. On the other hand, if the classical solution does not exist we apply a new concept of the so-called K convergence and show how to compute effectively the observable quantities of a space-time parametrized measure generated by numerical solutions. Consequently, we derive the strong convergence of the empirical averages of numerical solutions to a weak dissipative solution.