This is an announcement for the paper "On best proximity points in metric and Banach spaces" by Rafa Espinola and Aurora Fernandez-Leon. Abstract: 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. Archive classification: math.FA math.MG Mathematics Subject Classification: 54H25, 47H09 Remarks: 17 pages. Accepted for publication in the Canadian Mathematical The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0911.5263 or http://arXiv.org/abs/0911.5263 or by email in unzipped form by transmitting an empty message with subject line uget 0911.5263 or in gzipped form by using subject line get 0911.5263 to: math@arXiv.org.