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We estimate the concentration functions of n-fold convolutions of one-dimensional probability measures. The Kolmogorov-Rogozin inequality implies that for nondegenerate distributions these functions decrease at least as O(n(-1/2)). On the other hand, Esseen((3)) has shown that this rate is o(n(-1/2)) iff the distribution has an infinite second moment. This statement was sharpened by Morozova.((9)) Theorem 1 of this paper provides an improvement of Morozova's result. Moreover, we present more general estimates which imply the rates o(n(-1/2)).