  We compactify $M$-theory on a Calabi-Yau manifold to five dimensions by
wrapping the membrane and fivebrane solitons of the eleven-dimensional
supergravity limit around Calabi-Yau two-cycles and four-cycles respectively.
We identify the perturbative and non-perturbative BPS states thus obtained with
those of heterotic string theory compactified on $K3\times S^1$. Quantum
aspects of the five-dimensional theory are discussed.
