The large scale geometry of Gromov hyperbolic metric spaces exhibits many distinctive
features, such as the stability of quasi-geodesics (the Morse Lemma), the linear isoperimetric filling
inequality for 1-cycles, the visibility property, and the homeomorphism between visual boundaries
induced by a quasi-isometry. After briefly reviewing these properties, I will describe a number of closely
analogous results for spaces of rank n > 1 in an asymptotic sense, under some weak assumptions
reminiscent of non-positive curvature. A central role is played by a suitable class of n-dimensional
surfaces of polynomial growth of order n, which serve as a substitute for quasi-geodesics.