  Even linear operators on infinite-dimensional spaces can display interesting
dynamical properties and yield important links among functional analysis,
differential and global geometry and dynamical systems, with a wide range of
applications. In particular, hypercyclicity is an essentially
infinite-dimensional property, when iterations of the operator generate a dense
subspace. A Frechet space admits a hypercyclic operator if and only if it is
separable and infinite-dimensional. However, by considering the semigroups
generated by multiples of operators, it is possible to obtain hypercyclic
behaviour on finite dimensional spaces. This article gives a brief review of
some recent work on hypercyclicity of operators on Banach, Hilbert and Frechet
spaces.
