Exponential sums of large degree play a prominent role in the analysis of problems spanning the analytic theory of numbers. In 1935, I. M. Vinogradov devised a method for estimating their mean values very much more efficient than the methods available hitherto due to Weyl and van der Corput, and subsequently applied his new estimates to investigate the zero-free region of the Riemann zeta function, in Diophantine approximation, and in Waring?s problem. Recent applications from the 21st century include sum-product estimates in additive combinatorics, and the investigation of the geometry of moduli spaces. Over the past 75 years, estimates for the moments underlying Vinogradov?s mean value theorem have failed to achieve those conjectured by a factor of roughly log k in the number of implicit variables required to successfully analyse exponential sums of degree k. In this talk we will sketch out some history, several applications, and the ideas underlying our recent work which comes within a stone?s throw of the best possible conclusions.