We show that doubling at some large scale in a Cayley graph implies uniform doubling at all subsequent scales. The proof is based on the structure theorem for approximate subgroups proved by Green, Tao and the first author. We also give a number of applications to the geometry and spectrum of finite Cayley graphs. For example, we show that a finite group has moderate growth in the sense of Diaconis and Saloff-Coste if and only if its diameter is larger than a fixed power of the cardinality of the group. We call such groups almost flat and show that they have a subgroup of bounded index admitting a cyclic quotient of comparable diameter. We also give bounds on the Cheeger constant, first eigenvalue of the Laplacian, and mixing time. This can be seen as a finite-group version of Gromov's theorem on groups with polynomial growth. It also improves on a result of Lackenby regarding property (τ ) in towers of coverings. Another consequence is a universal upper bound on the diameter of all finite simple groups, independent of the CFSG. of cardinalities |S|, |S 2 |, |S 3 |, . . . , where we denote by S n the n-fold product set {s 1 · ... · s n ; s i ∈ S}. This is the ball of radius n in the Cayley graph of G relative to S. In the event that G is finite, we also consider the diameter diam S (G) of G with respect to S, which is defined to be the minimum n such that S n = G.According to Gromov's polynomial growth theorem [23], if the sequence {|S n |} is bounded above by a polynomial function of n, then G has a finite-index nilpotent subgroup. In this paper we are concerned