  Let ${X_1,...,X_n}$ be i.i.d. random observations. Let
$\mathbb{S}=\mathbb{L}+\mathbb{T}$ be a $U$-statistic of order $k\ge2$ where
$\mathbb{L}$ is a linear statistic having asymptotic normal distribution, and
$\mathbb{T}$ is a stochastically smaller statistic. We show that the rate of
convergence to normality for $\mathbb{S}$ can be simply expressed as the rate
of convergence to normality for the linear part $\mathbb{L}$ plus a correction
term, $(\operatorname {var}\mathbb{T})\ln^2(\operatorname {var}\mathbb{T})$,
under the condition ${\mathbb{E}\mathbb{T}^2<\infty}$. An optimal bound without
this $\log$ factor is obtained under a lower moment assumption ${\mathbb
{E}|\mathbb{T}|^{\alpha}<\infty}$ for ${\alpha<2}$. Some other related results
are also obtained in the paper. Our results extend, refine and yield a number
of related-known results in the literature.
