  We apply the efficient congruencing method to estimate Vinogradov's integral
for moments of order 2s, with 1<=s<=k^2-1. Thereby, we show that quasi-diagonal
behaviour holds when s=o(k^2), we obtain near-optimal estimates for
1<=s<=(1/4)k^2+k, and optimal estimates for s>=k^2-1. In this way we come half
way to proving the main conjecture in two different directions. There are
consequences for estimates of Weyl type, and in several allied applications.
Thus, for example, the anticipated asymptotic formula in Waring's problem is
established for sums of s kth powers of natural numbers whenever s>=2k^2-2k-8
(k>=6).
