  We initiate the study of correspondences for Smale spaces. Correspondences
are shown to provide a notion of a generalized morphism between Smale spaces
and are a special case of finite equivalences. Furthermore, for shifts of
finite type, a correspondence is related to a matrix which intertwines the
adjacency matrices of the shifts. This observation allows us to define an
equivalence relation on all Smale spaces which reduces to shift equivalence for
shifts of finite type. Several other notions of equivalence are introduced on
both correspondences and Smale spaces; a hierarchy between these equivalences
is established. Finally, we provide several methods for constructing
correspondences and provide specific examples.
