  Let $G$ be a multiplicative group, let $A,B \subseteq G$ be finite and
nonempty, and define the product set $AB = {ab \mid $a \in A$ and $b \in B$}$.
Two fundamental problems in combinatorial number theory are to find lower
bounds on $|AB|$, and then to determine structural properties of $A$ and $B$
under the assumption that $|AB|$ is small. We focus on the extreme case when
$|AB| < |A| + |B|$, and call any such pair $(A,B)$ \emph{critical}.
  In the case when $|G|$ is prime, the Cauchy-Davenport Theorem asserts that
$|AB| \ge \min {|G|, |A| + |B| - 1}$, and Vosper refined this result by
classifying all critical pairs in these groups. For abelian groups, Kneser
proved a natural generalization of Cauchy-Davenport by showing that there
exists $H \le G$ so that $|AB| \ge |A| + |B| - |H|$ and $ABH = AB$. Kemperman
then proved a result which characterizes the structure of all critical pairs in
abelian groups.
  Our main result gives a classification of all critical pairs in an arbitrary
group $G$. As a consequence of this we derive the following generalization of
Kneser's Theorem to arbitrary groups: There exists $H \le G$ so that $|AB| \ge
|A| + |B| - |H|$ and so that for every $y \in AB$ there exists $x \in G$ so
that $y(x^{-1} H x) \subseteq AB$.
