We construct shape spaces of elastic spherical surfaces immersed in Euclidean space Rk. The spaces are equipped with geodesic metrics that depend on the tension and rigidity of the surfaces. We develop algorithms to calculate geodesics and geodesic distances, as well as tools to quantify local shape similarities and contrasts, thus obtaining a local-global formulation. We give examples of geodesic interpolations and illustrations of the use of the model in brain mapping.