  We refine the classical Lindeberg-Feller central limit theorem by obtaining
asymptotic bounds on the Kolmogorov distance, the Wasserstein distance, and the
parametrized Prokhorov distances in terms of a Lindeberg index. We thus obtain
more general approximate central limit theorems, which roughly state that the
row-wise sums of a triangular array are approximately asymptotically normal if
the array approximately satisfies Lindeberg's condition. This allows us to
continue to provide information in non-standard settings in which the classical
central limit theorem fails to hold. Stein's method plays a key role in the
development of this theory.
