  In this paper we study the existence and uniqueness of best proximity points
of cyclic contractions as well as the convergence of iterates to such proximity
points. We do it from two different approaches, leading each one of them to
different results which complete, if not improve, other similar results in the
theory. Results in this paper stand for Banach spaces, geodesic metric spaces
and metric spaces. We also include an appendix on CAT(0) spaces where we study
the particular behavior of these spaces regarding the problems we are concerned
with.
