Our main research area is the study of the chaotic dynamics of linear operators on infinite dimensional spaces (Banach and Fréchet spaces). This is a topic which combines techniques of Dynamical Systems and Operator Theory. We are mainly concerned with the analysis of topological properties related to chaos, like the existence of dense orbits (hypercyclicity, within our framework), other stronger properties (mixing, weakly mixing, etc.), and general notions of irregularity for orbits (Li-Yorke chaos, distributional chaos,..). We also study notions from ergodic theory (given an infinite-dimensional linear dynamical system, the existence of invariant measures with certain additional properties,etc.) and their consequences in the frequently recurrent behaviour of dense orbits. We are interested in discrete and continuous dynamical systems. In the latter case, we mainly concentrate on continuous dynamical systems which come from solution semigroups of linear PDEs and of infinite systems of linear ODEs.