In hyperbolic geometry, a horoball is an object in hyperbolic n-space: the limit of a sequence of increasing balls sharing (on one side) a tangent hyperplane and its point of tangency. Its boundary is called a horosphere. For n = 2 a horosphere is called a horocycle.
This terminology is due to William Thurston, who used it in his work on hyperbolic 3-manifolds. Thus horosphere/horoball often has a connotation of referring to 3-dimensional hyperbolic geometry.
In the conformal ball model, a horoball is represented by a ball whose surface is tangent to the horizon sphere. In the upper half-space model, a horoball can appear as such a sphere, or as a half-space whose lower boundary is parallel to the horizon plane. In the hyperboloid model, a horoball is the region above a plane whose normal lies in the asymptotic cone.
A horosphere has a critical amount of (isotropic) curvature: if the curvature were any greater, the surface would be able to close, yielding a sphere, and if the curvature were any less, the surface would be an N-1 dimensional hypercycle (a hyperhypercycle).