  Motivated by a recent investigation of Costakis et al. on the notion of
recurrence in linear dynamics, we study various stronger forms of recurrence
for linear operators, in particular that of frequent recurrence. We study,
among other things, the relationship between a type of recurrence and the
corresponding notion of hypercyclicity, the influence of power boundedness, and
the interplay between recurrence and spectral properties. We obtain, in
particular, Ansari- and L\'eon-M\"uller-type theorems for
$\mathcal{F}$-recurrence under very weak assumptions on the Furstenberg family
$\mathcal{F}$. This allows us, as a by-product, to deduce Ansari- and
L\'eon-M\"uller-type theorems for $\mathcal{F}$-hypercyclicity.
