For $\alpha$ an ordinal, we investigate the class $\mathscr{SZ}_\alpha$
consisting of all operators whose Szlenk index is an ordinal not exceeding
$\omega^\alpha$. Our main result is that $\mathscr{SZ}_\alpha$ is a closed,
injective, surjective operator ideal for each $\alpha$. We also study the
relationship between the classes $\mathscr{SZ}_\alpha$ and several well-known
closed operator ideals.