Hyperbolic systems of conservation laws are known to produce discontinuous solutions in finite time. These discontinuous solutions to a differential equation can be only carried on in the context of weak solutions. Weak solutions are in general non-unique and additional criterions are needed to only allow physically relevant solutions. Recent results show that classical entropy inequalities, i.e. the classical notion of entropy for equilibrium thermodynamics, are not sufficient. Dafermos proposed the entropy rate criterion as an alternative criterion to select weak solutions. A weak solution satisfying this criterion should dissipate entropy as fast or faster than all other weak solutions. In this book Finite-Volume and Discontinuous Galerkin methods are presented that enforce this entropy rate criterion for numerical solutions. Key to these schemes is the prediction of the maximal possible entropy dissipation by an exact weak solution. This entropy decay is afterwards enforced for the approximate weak solutions calculated by the numerical schemes. The new schemes show essentially non-oscillatory, robust and stable behavior over a wide range of testcases. The tests used range from one-dimensional scalar conservation laws to transonic and supersonic solutions to the full Euler equations on unstructured meshes.