  We study multiply recurrent and hypercyclic operators as a special case of
$\mathcal F$-hypercyclicity, where $\mathcal F$ is the family of subsets of the
natural numbers containing arbitrarily long arithmetic progressions. We prove
several properties of hypercyclic multiply recurrent operators, we characterize
those operators which are weakly mixing and multiply recurrent, and we show
that there are operators that are multiply recurrent and hypercyclic but not
weakly mixing.
