  We derive a new representation for $U$- and $V$-statistics. Using this
representation, the asymptotic distribution of $U$- and $V$-statistics can be
derived by a direct application of the Continuous Mapping theorem. That novel
approach not only encompasses most of the results on the asymptotic
distribution known in literature, but also allows for the first time a unifying
treatment of non-degenerate and degenerate $U$- and $V$-statistics. Moreover,
it yields a new and powerful tool to derive the asymptotic distribution of very
general $U$- and $V$-statistics based on long-memory sequences. This will be
exemplified by several astonishing examples. In particular, we shall present
examples where weak convergence of $U$- or $V$-statistics occurs at the rate
$a_n^3$ and $a_n^4$, respectively, when $a_n$ is the rate of weak convergence
of the empirical process. We also introduce the notion of asymptotic (non-)
degeneracy which often appears in the presence of long-memory sequences.
