  Using ``Tate's algorithm,'' we identify loci in the moduli of F-theory
compactifications corresponding to enhanced gauge symmetry. We apply this to
test the proposed F-theory/heterotic dualities in six dimensions. We recover
the perturbative gauge symmetry enhancements of the heterotic side and the
physics of small $SO(32)$ instantons, and discover new mixed
perturbative/non-perturbative gauge symmetry enhancements. Upon further
toroidal compactification to 4 dimensions, we derive the chain of Calabi-Yau
threefolds dual to various Coulomb branches of heterotic strings.
