We present a summary of the main lines of argument in the new method for estimating exponential sums introduced by Bombieri and Iwaniec in 1986. The key ideas are to dissect the exponential sum into sections corrésponding to the rational numbers of bounded height, and to bound these subsums in the mean using the large sieve. Applications include bounds for the Riemann zeta function, both mean and pointwise, and the number of integer points in a plane domain with smooth convex boundary. Some problems are posed on the Diophantine bebaviour of a smooth curve at integer arguments. The bibliography lists all papers related to the new method up to the end of 1989.