  Let $(X, \cal B, \nu)$ be a probability space and let $\Gamma$ be a countable
group of $\nu$-preserving invertible maps of $X$ into itself. To a probability
measure $\mu$ on $\Gamma$ corresponds a random walk on $X$ with Markov operator
$P$ given by $P\psi(x) = \sum_{a} \psi(ax) \, \mu(a)$. A powerful tool is the
spectral gap property for the operator $P$ when it holds. We consider various
examples of ergodic $\Gamma$-actions and random walks and their extensions by a
vector space: groups of automorphisms or affine transformations on compact
nilmanifolds, random walk in random scenery on non amenable groups,
translations on homogeneous spaces of simple Lie groups, random walks on motion
groups. The spectral gap property is applied to obtain limit theorems,
recurrence/transience property and ergodicity for random walks on non compact
extensions of the corresponding dynamical systems.
