  In this paper we consider closed orientable surfaces $M$ of positive genus
and diffeomorphisms $f:M\rightarrow M.$ The main objective is to show that such
maps, under certain natural conditions and some generic assumptions, can have
unbounded periodic open disks, which are very weird objects. For instance, if
$D\subset M$ is such a disk (a maximal one), then the connected components of
its lift to the universal cover of $M,$ denoted $\widetilde{M},$ all share the
same boundary $K,$ which is the lift of $\partial D$ to $\widetilde{M}.$ In
other words, there is an equivariant closed connected set $K$ in
$\widetilde{M}$ (in particular, $K$ is unbounded in every direction), which is
equal to the boundary of all connected components of the lift of $D.$
  We present two main results. The first describes the dynamics of $f\mid _D$
when the rotation number of the prime ends compactification of $D$ is rational.
It implies in particular that $D$ is either the basin of an attractor or of a
repeller contained in $D$. The other result shows that unbounded periodic open
disks appear for generic one-parameter families $f_t:M\rightarrow M$ whenever
the rotation set of $f_t$ grows.
  When $M$ is the torus, apart from some $C^r$-generic conditions (which hold
for all $r\geq 1),$ we also assume that the rotation set of $f$ $($or $f_t)$
has non-empty interior. For higher genus surfaces, we need a more complicated
hypothesis, the existence of a fully essential system of curves.
