  A $U$-statistic of a Poisson point process is defined as the sum $\sum
f(x_1,\ldots,x_k)$ over all (possibly infinitely many) $k$-tuples of distinct
points of the point process. Using the Malliavin calculus, the Wiener-It\^{o}
chaos expansion of such a functional is computed and used to derive a formula
for the variance. Central limit theorems for $U$-statistics of Poisson point
processes are shown, with explicit bounds for the Wasserstein distance to a
Gaussian random variable. As applications, the intersection process of Poisson
hyperplanes and the length of a random geometric graph are investigated.
