  In six dimensions, cancellation of gauge, gravitational, and mixed anomalies
strongly constrains the set of quantum field theories which can be coupled
consistently to gravity. We show that for some classes of six-dimensional
supersymmetric gauge theories coupled to gravity, the anomaly cancellation
conditions are equivalent to tadpole cancellation and other constraints on the
matter content of heterotic/type I compactifications on K3. In these cases, all
consistent 6D supergravity theories have a realization in string theory. We
find one example which may arise from a novel string compactification, and we
identify a new infinite family of models satisfying anomaly factorization. We
find, however, that this infinite family of models, as well as other infinite
families of models previously identified by Schwarz are pathological. We
suggest that it may be feasible to demonstrate that there is a string theoretic
realization of all consistent six-dimensional supergravity theories which have
Lagrangian descriptions with arbitrary gauge and matter content. We attempt to
frame this hypothesis of string universality as a concrete conjecture.
