For the incompressible Euler equations, Onsager's Conjecture (which, meanwhile, can be considered a theorem) tells that a solution will conserve energy if its regularity is above a certain threshold. The conjecture was originally motivated from Kolmogorov's theory of turbulence. While the proof of this statement on the whole space is by now classical, recently there has been much activity to generalize or improve the theory to deal with questions related to general conservation laws, boundary effects, critical function spaces, and degenerate situations like vacuum formation in compressible fluids. The talk will be somewhat complementary to the one of Claude Bardos earlier in this seminar series.