Ordinal indices have been used in several ways to quantify behaviors and structures in infinite dimensional Banach spaces. In this talk I will survey some results which make use of ordinal indices. Among those indices discussed will be the Szlenk index, Bourgain's $\ell_1$ index, and the $\ell_1^+$ weakly null index. Along the way I will define the Schreier sets and indicate the role that they have played in the theory. While the talk will be far from exhaustive, I hope that the selection will give non-experts in Banach space theory a useful introduction.