This paper deals with the definition and properties of the velocity space of special or general relativity. The metric naturally induced by the enveloping space-time is that of a space of constant negative curvature which is referred to here as a 3-dimensional Lobachevsky space. Our purpose is to show how this notion can simplify or reinterpret ideas of relativistic kinematics. Supplementing the more familiar applications, three others are given here which make essential use of the curvature, namely to the derivations of aberration, Thomas precession, and the forms of relativistic collision invariants.