  Some basic notions and results in Topological Dynamics are extended to
continuous groupoid actions in topological spaces. We focus mainly on
recurrence properties. Besides results that are analogous to the classical case
of group actions, but which have to be put in the right setting, there are also
new phenomena. Mostly for groupoids whose source map is not open (and there are
many), some properties which were equivalent for group actions become distinct
in this general framework; we illustrate this with various counterexamples.
