Partition monoids and algebras arise in representation theory in the context of Schur-Weyl duality in symmetric groups. They are also interesting from a semigroup theoretic point of view as they contain the full transformation semigroups as well as the symmetric (and dual symmetric) inverse monoids. We characterise the elements of the partition monoid which are products of idempotent partitions. The finite and infinite cases require separate treatment. Our results are similar in character to those obtained by Howie in his 1966 paper on full transformation semigroups; in fact, Howie's results may also be deduced from ours. This is joint work with Des FitzGerald.