I will discuss joint work with Rob Morris on the issue of counting the number of sets A in Z/NZ of a given size with doubling constant at most K. I will also mention some applications, for example to the problem of estimating the probability that if S is a random set of natural numbers then all except k natural numbers are the sum of two elements of S, and to the problem of finding an asymptotic for the clique number of a random Cayley graph on Z/NZ.