A new formulation of nonrelativistic scattering theory is developed which expresses the S matrix as a path integral. This formulation appears to have at least two advantages: (1) A closed formula is obtained for the S matrix in terms of the potential, not involving a series expansion; (2) the energy-conserving δ function can be explicitly extracted using a technique analogous to that of Faddeev and Popov, thereby yielding a closed path-integral expression for the T matrix. The introduction of the concept of the classical interaction picture provides considerable physical insight into this formulation. Finally, this formulation also suggests a succession of improvements to the eikonal approximation, the first of which is discussed explicitly.