Digital computer performance speed has increased by about eight orders of magnitude since computers were invented in the 1930s. However, such large, complex problems as high-resolution image processing and models of atmospheric behavior will require even faster operation. This paper estimates potential computer speeds, identifies the principles of physics that will ultimately limit them, and discusses the social value of supercomputers. Theoretically, speed could be significantly increased by representing numbers in remainder or residue form rather than in conventional positional notation, but practical considerations hinder such procedures. CPU utilization, typically about 50 percent to 60 percent, could be increased with multistream or pipeline processing. The small size and dense packaging necessary to fast computers are inconsistent with heat dissipation, but the size and density limits have not been even approached. Computers 1000 or even 10,000 times faster than the best machines today are possible. Such a machine would be of enormous value to society. 21 pp. Ref.