In this article, we consider a topological dynamical system. The generic
limit set is the smallest closed subset which has a comeager realm of
attraction. We study some of its topological properties, and the links with
equicontinuity and sensitivity. We emphasize the case of cellular automata, for
which the generic limit set is included in all subshift attractors, and discuss
directional dynamics, as well as the link with measure-theoretical similar
notions