  An $integral$ of a group $G$ is a group $H$ whose derived group (commutator
subgroup) is isomorphic to $G$. This paper discusses integrals of groups, and
in particular questions about which groups have integrals and how big or small
those integrals can be. Our main results are:
  (1) If a finite group has an integral, then it has a finite integral.
  (2) A precise characterization of the set of natural numbers $n$ for which
every group of order $n$ is integrable: these are the cubefree numbers $n$
which do not have prime divisors $p$ and $q$ with $q\mid p-1$.
  (3) An abelian group of order $n$ has an integral of order at most
$n^{1+o(1)}$, but may fail to have an integral of order bounded by $cn$ for
constant $c$.
  (4) A finite group can be integrated $n$ times (in the class of finite
groups) if and only if it is the central product of an abelian group and a
perfect group.
  There are many other results on such topics as centreless groups, groups with
composition length $2$, and infinite groups. We also include a number of open
problems.
