A compact non-imaging lens is described and analyzed: the Totally Internally Reflecting (TIR) lens. It constitutes a major class of optical devices distinct from reflectors and Fresnel lenses. It is a transmissive device that redirects light passing through it via the action of a multiplicity of prismatic facets basically acting as annular Harting-Dove prisms that rotationally `wash-out' image structure. They can achieve much larger bend angles (well over 90 degree(s)) than those of the refraction-only facets of Fresnel lenses. As a consequence, TIR lenses are extremely compact, typically having a thickness about one fifth their diameter. This paper discusses their applications as collimators for small light sources (LEDs and HID lamps), injectors for fiber- optic illumination systems, and solar concentrators, and how close their performance comes to the ideal thermodynamic limit.© (1991) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.