For the integers $a$ and $b$ let $P(a^b)$ be all partitions of the
set $N= {1,..., ab}$ into parts of size $a.$ Further, let
$\mathbb{C}P (a^b)$ be the corresponding permutation module for the
symmetric group acting on $N.$ A conjecture of Foulkes says
that $\mathbb{C}P (a^b)$ is isomorphic to a submodule of $\mathbb{C}P
(b^a)$ for all $a$ not larger than $b.$ The conjecture goes back to
the 1950's but has remained open. Nevertheless, for some values of
$b$ there has been progress. I will discuss some proofs and further
conjectures. There is a close correspondence between the
representations of the symmetric groups and those of the general
linear groups, via Schur-Weyl duality. Foulkes' conjecture therefore
has implications for $GL$-representations. There are interesting
connections to classical invariant theory which I hope to mention.