  We investigate the operation of torus surgery on tori embedded in $S^4$. Key
questions include which 4-manifolds can be obtained in this way, and the
uniqueness of such descriptions. As an application we construct embeddings of
3-manifolds into 4-manifolds by viewing Dehn surgery as a cross section of a
surgery on a surface. In particular, we give new embeddings of homology spheres
into $S^4$.
