An atomic-gas Bose-Einstein Condensate, placed in the periodic light-shift potential of an optical standing wave, exhibits many features that are similar to the familiar problem of electrons moving in the periodic potential of a solid-state crystal lattice. Among the differences are that the atoms in an optical lattice offer possibilities for measurement and control that are not readily available in traditional condensed systems. Experiments that are difficult or impossible with solids are often straightforward with optical lattices, sometimes with surprising results. Among these are single-particle effects like Bloch oscillations and many-body problems like the Hubbard Model and the Mott insulator transition. Cold atomic gases hold the promise of being quantum simulators of calculationally intractable problems of both fundamental and practical interest.