We provide an overview of efficient congruencing as applied to mean values asociated with translation-dilation invariant (TDI) systems of equations allied to that of Vinogradov. This should offer some insight into the ideas underlying the basic version of the method, and some sense of the scope of the workshop. We should have time to sketch how to adapt the method to TDI families of systems, and say something about the main conjecture in several settings.

Some of this work was supported by a European Research Council Advanced Grant under the European Union's Horizon 2020 research and innovation programme under grant agreement No. 695223.