  In this paper we study the global structure of the stable homotopy theory of
spectra. We establish criteria for when the homotopy theory associated to a
given stable model category agrees with the classical stable homotopy theory of
spectra. One sufficient condition is that the associated homotopy category is
equivalent to the stable homotopy category as a triangulated category with an
action of the ring of stable homotopy groups of spheres. In other words, the
classical stable homotopy theory, with all of its higher order information, is
determined by the homotopy category as a triangulated category with an action
of the stable homotopy groups of spheres. Another sufficient condition is the
existence of a small generating object (corresponding to the sphere spectrum)
for which a specific `unit map' from the infinite loop space QS^0 to the
endomorphism space is a weak equivalence.
