  In this paper, we introduce and study the notion of super-recurrence of
operators. We investigate some properties of this class of operators and show
that it shares some characteristics with supercyclic and recurrent operators.
In particular, we show that if $T$ is super-recurrent, then $\sigma(T)$ and
$\sigma_p(T^*)$, the spectrum of $T$ and the point spectrum of $T^*$
respectively, have some noteworthy properties.
