  We give some rates of convergence in the distances of Kolmogorov and
Wasserstein for standardized martingales with differences having finite
variances. For the Kolmogorov distances, we present some exact Berry-Esseen
bounds for martingales, which generalizes some Berry-Esseen bounds due to
Bolthausen. For the Wasserstein distance, with Stein's method and Lindeberg's
telescoping sum argument, the rates of convergence in martingale central limit
theorems recover the classical rates for sums of i.i.d.\ random variables, and
therefore they are believed to be optimal.
