Uniform and nonuniform Berry--Esseen (BE) bounds of optimal orders on the closeness to normality for general nonlinear statistics are obtained. Applications to Student's, Pearson's, and Hotelling's statistics are given, which appear to be the first known results of these kinds (with the exception of uniform BE bounds for Student's statistic). When applied to the well-studied case of the central Student statistic, our general results compare well with known ones in that case, obtained previously by specialized methods. The proofs use a Stein-type method developed by Chen and Shao, a Cram\'er-type of tilt transform, exponential and Rosenthal-type inequalities for sums of random vectors established by Pinelis, Sakhanenko, and Utev, as well as a number of other, quite recent results motivated by this study. The method allows one to obtain bounds with explicit and rather moderate-size constants, at least as far as the uniform bounds are concerned.
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