  Recent developments in the theory and application of the Hardy-Littlewood
method are discussed, concentrating on aspects associated with diagonal
diophantine problems. Recent efficient differencing methods for estimating mean
values of exponential sums are described first, concentrating on developments
involving smooth Weyl sums. Next, arithmetic variants of classical inequalities
of Bessel and Cauchy-Schwarz are discussed. Finally, some emerging connections
between the circle method and arithmetic geometry are mentioned.
