  To every Gromov hyperbolic space X one can associate a space at infinity
called the Gromov boundary of X. Gromov showed that quasi-isometries of
hyperbolic metric spaces induce homeomorphisms on their boundaries, thus giving
rise to a well-defined notion of the boundary of a hyperbolic group. Croke and
Kleiner showed that the visual boundary of non-positively curved (CAT(0))
groups is not well-defined, since quasi-isometric CAT(0) spaces can have
non-homeomorphic boundaries. For any sublinear function $\kappa$, we consider a
subset of the visual boundary called the $\kappa$-Morse boundary and show that
it is QI-invariant and metrizable. This is to say, the $\kappa$-Morse boundary
of a CAT(0) group is well-defined. In the case of Right-angled Artin groups, it
is shown in the Appendix that the Poisson boundary of random walks is naturally
identified with the $\sqrt{t \log t}$--boundary.
