Abstract: A natural notion of energy for a map is given by measuring how much the map stretches at each point and integrating that quantity over the domain. Harmonic maps are critical points for the energy and existence and compactness results for harmonic maps have played a major role in the advancement of geometric analysis. Gromov-Schoen and Korevaar-Schoen developed a theory of harmonic maps into metric spaces with non-positive curvature in order to address rigidity problems in geometric group theory. In this talk we discuss harmonic maps into CAT(k) spaces which are metric spaces with positive upper curvature bounds. By proving global existence and analyzing the local behavior of such maps, we determine a uniformization theorem for CAT(k) spheres.