  For $\alpha$ an ordinal, we investigate the class $\mathscr{SZ}_\alpha$
consisting of all operators whose Szlenk index is an ordinal not exceeding
$\omega^\alpha$. We show that each class $\mathscr{SZ}_\alpha$ is a closed
operator ideal and study various operator ideal properties for these classes.
The relationship between the classes $\mathscr{SZ}_\alpha$ and several
well-known closed operator ideals is investigated and quantitative
factorization results in terms of the Szlenk index are obtained for the class
of Asplund operators.
