Since the celebrated Kolmogorov Theory of Turbulence from 1941 it has been clear that, at first approximation, a good understanding of incompressible turbulence is subject to the study of weak solutions to the Euler equations having critical regularity. After presenting the problem, and propose a definition of "critical" regularity, I will show how to prove a formula for the anomalous dissipation measure which improves the previously known ones. This is then used to prove interior local energy conservation for bounded solutions with bounded variation, building on measure theoretic ideas introduced by Luigi Ambrosio in the context of the Diperna-Lions theory for linear transport equations. If time permits, I will then show how to include physical boundaries in the above analysis. The geometry of both the boundary and the velocity here plays a major role, and understanding how the velocity tends to be tangential to the boundary is necessary to allow any Lipschitz domain.