  We introduce and study the Bourgain index of an operator between two Banach
spaces. In particular, we study the Bourgain $\ell_p$ and $c_0$ indices of an
operator. Several estimates for finite and infinite direct sums are
established. We define classes determined by these indices and show that some
of these classes form operator ideals. We characterize the ordinals which occur
as the index of an operator and establish exactly when the defined classes are
closed. We study associated indices for non-preservation of $\ell_p^\xi$ and
$c_0^\xi$ spreading models and indices characterizing weak compactness of
operators between separable Banach spaces. We also show that some of these
classes are operator ideals and discuss closedness and distinctness of these
classes.
