The Euler equations of fluid dynamics is the primary example of a system of hyperbolic conservation laws. Such systems are well-posed in one space dimension under a "smallness" assumption, but for realistic initial data in multiple dimensions there is a great lack of stability, existence and uniqueness theory. Moreover, certain initial-value problems are indeed unstable with respect to initial data. These facts indicate an inherent uncertainty in the solution, even when the initial data is given exactly. We advocate the point of view of so-called measure-valued solutions, and give numerical evidence that this might be the correct notion of solutions for hyperbolic conservation laws. We prove the existence and stability of measure-valued solutions in certain special cases, and design numerical algorithms that show stable, convergent behavior in unstable initial-value problems.