We begin by outlining the circle method and applications such as Waring's problem, before considering the classical approach to Vinogradov's mean value theorem (VMVT). This mean value theorem for exponential sums over the integers can be generalised to other settings, such as function fields, as we outline. We then describe the efficient congruencing approach to VMVT, which can be seen as a p-adic concentration argument exploiting the translation-invariance of the underlying Diophantine system. This circle of ideas yields novel conclusions concerning the solutions of congruences in many variables loosely analogous to Hensel's lemma, as well as conclusions of interest in discrete restriction theory. Refinements provide a proof of the main conjecture concerning VMVT. The recent work of Bourgain, Demeter and Guth on decoupling and VMVT may be seen as a real analogue of this (p-adic) efficient congruencing method.