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We give some rates of convergence in the distances of Kolmogorov andWasserstein for standardized martingales with differences having finitevariances. For the Kolmogorov distances, we present some exact Berry-Esseenbounds for martingales, which generalizes some Berry-Esseen bounds due toBolthausen. For the Wasserstein distance, with Stein's method and Lindeberg'stelescoping sum argument, the rates of convergence in martingale central limittheorems recover the classical rates for sums of i.i.d.\ random variables, andtherefore they are believed to be optimal.

Rates of convergence in the distances of Kolmogorov and Wasserstein for standardized martingales