  We establish an exponential inequality for degenerated $U$-statistics of
order $r$ of i.i.d. data. This inequality gives a control of the tail of the
maxima absolute values of the $U$-statistic by the sum of two terms: an
exponential term and one involving the tail of $h\left(X_1,\dots,X_r\right)$.
We also give a version for not necessarily degenerated $U$-statistics having a
symmetric kernel and furnish an application to the convergence rates in the
Marcinkiewicz law of large numbers. Application to invariance principle in
H\"older spaces is also considered.
