We introduce a prime end-type theory on complete Kobayashi hyperbolic manifolds using horo-sphere sequences. This allows to introduce a new notion of boundary—new even in the unit disc in the complex space—the horosphere boundary, and a topology on the manifold together with its horosphere boundary, the horosphere topology. We prove that a bounded strongly pseudoconvex domain endowed with the horosphere topology is homeomorphic to its Euclidean closure, while for the polydisc such a horosphere topology is not even Hausdorff. We use this theory to study boundary behavior of univalent maps from bounded strongly pseudoconvex domains. Among other things, we prove that every univalent map of the unit ball whose image is bounded and convex, extends as a homeomorphism up to the closure. We also compare the horosphere boundary with the Gromov boundary in the special case of the bidisc, proving that they are not homeomorphic.