SO(8) and G_2 X G_2 N=1 Supergravities in D=6

We construct the N=1 supergravity analog of the Green-Schwarz and heterotic superstring theories in 10D. We find the SO(8) theory previously found by compactification of 10D on $K3$. We also find eleven $G_2\times G_2$ theories, two with symmetric matter content. It is not obvious how ten of the $G_2\times G_2$ theories can be gotten from 10D.