We introduce two new classes of fusion categories which are obtained by a
certain procedure from finite groups - weakly group-theoretical categories and
solvable categories. These are fusion categories that are Morita equivalent to
iterated extensions (in the world of fusion categories) of arbitrary,
respectively solvable finite groups. Weakly group-theoretical categories have
integer dimension, and all known fusion categories of integer dimension are
weakly group theoretical. Our main results are that a weakly group-theoretical
category C has the strong Frobenius property (i.e., the dimension of any simple
object in an indecomposable C-module category divides the dimension of C), and
that any fusion category whose dimension has at most two prime divisors is
solvable (a categorical analog of Burnside's theorem for finite groups). This
has powerful applications to classification of fusion categories and
semsisimple Hopf algebras of a given dimension. In particular, we show that any
fusion category of integer dimension <84 is weakly group-theoretical (i.e.
comes from finite group theory), and give a full classification of semisimple
Hopf algebras of dimensions pqr and pq^2, where p,q,r are distinct primes.
