  Uniform and nonuniform Berry--Esseen (BE) bounds of optimal orders on the
closeness to normality for general abstract nonlinear statistics are given,
which are then used to obtain optimal bounds on the rate of convergence in the
delta method for vector statistics. Specific applications to Pearson's,
non-central Student's and Hotelling's statistics, sphericity test statistics, a
regularized canonical correlation, and maximum likelihood estimators (MLEs) are
given; all these uniform and nonuniform BE bounds appear to be the first known
results of these kinds, except for uniform BE bounds for MLEs. 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. For instance,
one has the uniform BE bound
$3.61\mathbb{E}(Y_1^6+Z_1^6)\,(1+\sigma^{-3})/\sqrt n$ for the Pearson sample
correlation coefficient based on independent identically distributed random
pairs $(Y_1,Z_1),\dots,(Y_n,Z_n)$ with $\mathbb{E} Y_1=\mathbb{E}
Z_1=\mathbb{E} Y_1Z_1=0$ and $\mathbb{E} Y_1^2=\mathbb{E} Z_1^2=1$, where
$\sigma:=\sqrt{\mathbb{E} Y_1^2Z_1^2}$.
