  We show the existence of global gauge anomalies in six dimensions for gauge
groups SU(2),SU(3) and G_2 coupled to matter, characterized by an element of
Z_{12},Z_6 and Z_3 respectively. Consideration of this anomaly rules out some
of the recently proposed 6 dimensional N=1 QFT's which were conjectured to
possess IR fixed point at infinite coupling. We geometrically engineer
essentially all the other models with one tensor multiplet using F-theory. In
addition we construct 3 infinite series using F-theory geometry which do not
have field theory analogs. All these models in the maximally Higgsed phase
correspond to the strong coupling behaviour of E_8 x E_8 heterotic string
compactification on K3 with instanton numbers (12+n,12-n).
