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Over the past 25 years (1979-2004), the theoretical-particle physicists who practice lattice-gauge theory have become some of the most frequent users of supercomputing cycles. Although we've long known that atoms consist of electrons surrounding a nucleus made of nucleons (protons and neutrons), we've recently learned that at a deeper level, the nucleons themselves are composites. We can best explain the strong forces between them by assuming they are composed of three quarks interacting via fields called gluons. The need for three constituents helps explain much of the zoo of similar states seen in particle physics experiments. Certain intractable aspects of the interactions between quarks and gluons have driven us to the computer. Indeed, large-scale simulations have helped us make major inroads into issues highly resistant to traditional approaches. Lattice-gauge theory provides a controlled scheme for studying strong interactions at low energies. The article shows that the main tools are powerful but demanding algorithms (such as conjugate-gradient sparse matrix inversions). Still-unsolved issues involve the "sign" problem and the basic formulation of parity violation on the lattice. Although we certainly need additional computing capability, we also need new ideas.