  Let K >= 1 be a parameter. A K-approximate group is a finite set A in a
(local) group which contains the identity, is symmetric, and such that A^2 is
covered by K left translates of A.
  The main result of this paper is a qualitative description of approximate
groups as being essentially finite-by-nilpotent, answering a conjecture of H.
Helfgott and E. Lindenstrauss. This may be viewed as a generalisation of the
Freiman-Ruzsa theorem on sets of small doubling in the integers to arbitrary
groups.
  We begin by establishing a correspondence principle between approximate
groups and locally compact (local) groups that allows us to recover many
results recently established in a fundamental paper of Hrushovski. In
particular we establish that approximate groups can be approximately modeled by
Lie groups.
  To prove our main theorem we apply some additional arguments essentially due
to Gleason. These arose in the solution of Hilbert's fifth problem in the
1950s.
  Applications of our main theorem include a finitary refinement of Gromov's
theorem, as well as a generalized Margulis lemma conjectured by Gromov and a
result on the virtual nilpotence of the fundamental group of Ricci almost
nonnegatively curved manifolds.
