In 1953, Richard Feynman showed how superfluidity can be understood from the point of view of imaginary-time path integrals. In this picture, each atom becomes a cyclic polymer and the superfluid transition occurs when the atomic paths join up to make larger units which stretch a macroscopic distance. Boson statistics imply that all possible ways of connecting or "permuting" the "polymers" have equal weight. Feynman's path integrals are an exact isomorphism between quantum statistical mechanics of bosons and the classical statistical mechanics of these special "polymers". We have implemented Feynman's path integral method on a computer. Efficient methods, based on the Metropolis algorithm have been developed to move the paths quickly through path configuration space and permutation space. In addition, accurate approximations to the many-body off-diagonal density matrix involving pairs and triplets of atoms have been calculated.