III Luminosity 7 IV Beam - Beam Interaction 8 V Crossing Angle and Crab Crossing 10 VI Optical Aberrations 12

In colliding beam facilities, the ""nal focus system" must demagnify the beams to attain the very small spot sizes required at the interaction points. The rst nal focus system with local chromatic correction was developed for the Stanford Linear Collider where very large demagniications were desired. This same conceptual design has been adopted by all the future linear col-lider designs as well as the SuperConducting Supercollider, the Stanford and KEK B-Factories, and the proposed Muon Collider. In this paper, the overall layout, physics constraints, and optimization techniques relevant to the design of nal-focus systems for high-energy electron-positron linear colliders are reviewed. Finally, advanced concepts to avoid some of the limitations of these systems are discussed. Abstract In colliding beam facilities, the ""nal focus system" must demagnify the beams to attain the very small spot sizes required at the interaction points. The rst nal focus system with local chromatic correction was developed for the Stanford Linear Collider where very large demagniications were desired. This same conceptual design has been adopted by all the future linear collider designs as well as the SuperConducting Supercollider, the Stanford and KEK B-Factories, and the proposed Muon Collider. In this paper, the overall layout , physics constraints, and optimization techniques relevant to the design of nal-focus systems for high-energy electron-positron linear colliders are reviewed. Finally, advanced concepts to avoid some of the limitations of these systems are discussed.