The Onsager conjecture, recently solved by Phil Isett, states that, below a certain threshold regularity, Hoelder continuous solutions of the Euler equations might dissipate the kinetic energy. The original work of Onsager was motivated by the phenomenon of anomalous dissipation and a rigorous mathematical justification of the latter should show that the energy dissipation in the Navier-Stokes equations is, in a suitable statistical sense, independent of the viscosity. In particular it makes much more sense to look for solutions of the Euler equations which, besides dissipating the {\em total} kinetic energy, satisfy as well a suitable form of local energy inequality. Such solutions were first shown to exist by Laszlo Szekelyhidi Jr. and myself. In this talk I will review the methods used so far to approach their existence and the most recent results by Isett and by Hyunju Kwon and myself.

In these three lectures I will give an overview of the DiPerna-Lions theory for transport equations and ODEs with Sobolev coefficients, including the developments of the last two decades, namely Ambrosio's extension to the case of BV coefficients, Lagrangian estimates and Bressan's mixing conjecture, near incompressibility of the flow versus bounds on the divergence of the field, the role of summability and the impact of convex integration techniques. I will try to cover the general ideas of several results, without going into the technical details, and I will give a special emphasis on open problems.