The main goal of this paper is the investigation of a relevant
property which appears in the various definition of deterministic
topological chaos for discrete time dynamical system:
transitivity. Starting from the standard Devaney's notion of topological chaos
based on regularity, transitivity, and sensitivity to the initial
conditions, the critique formulated by Knudsen is taken into
account in order to exclude periodic chaos from this definition.
Transitivity (or some stronger versions of it) turns out to be the
relevant condition of chaos and its role is discussed by a survey
of some important results about it with the presentation of some
new results. In particular, we study topological mixing, strong transitivity,
and full transitivity. Their applications to symbolic dynamics are
investigated with respect to the relationships with the associated
languages.