We provide bounds on the Wasserstein distance between the distribution of a degenerate, not necessarily symmetric U-statistic of independent random variables and the standard normal distribution. One main consequence of these bounds is a complete quantitative counterpart to a theorem by P. de Jong from 1990 which states that, under a certain negligibility condition, a sequence of such U-statistics satisfies a CLT whenever the sequence of fourth moments converges to 3.

We will also discuss approximation by a centered Gamma distribution and a generalization of de Jong's theorem to the multivariate case of vectors of such U-statistics.

Finally, if time allows, then we will also address the classical case of symmetric U-statistics.