  We develop a new quantitative approach to a multidimensional version of the
well-known {\it de Jong's central limit theorem} under optimal conditions,
stating that a sequence of Hoeffding degenerate $U$-statistics whose fourth
cumulants converge to zero satisfies a CLT, as soon as a Lindeberg-Feller type
condition is verified. Our approach allows one to deduce explicit (and
presumably optimal) Berry-Esseen bounds in the case of general $U$-statistics
of arbitrary order $d\geq1$. One of our main findings is that, for vectors of
$U$-statistics satisfying de Jong' s conditions and whose covariances admit a
limit, componentwise convergence systematically implies joint convergence to
Gaussian: this is the first instance in which such a phenomenon is described
outside the frameworks of homogeneous chaoses and of diffusive Markov
semigroups.
