Abstract: The Gromov boundary, defined for a \delta-hyperbolic metric space, plays a central role in many aspects of geometric group theory. We develop an analogous theory of boundary when the condition on hyperbolicity is removed: For a given proper, geodesic metric space X and a given sublinear function \kappa, we define the \kappa-boundary, as the space of all \kappa-Morse quasi-geodesics rays. We show that this boundary is QI-invariant and thus can be associated to any finitely generated group. For a large class of groups, we also show that the \kappa-boundary is large, that is, it provides a topological model for the Poisson boundary of the group. The talk is based on several joint projects with Ilya Gekhtman, Yulan Qing and Giulio Tiozzo.