  We discuss our attempts to generalize the known examples of dualities in N=1
supersymmetric gauge theories to exceptional gauge groups. We derive some dual
pairs from known examples connected to exceptional groups and find an
interesting phenomenon: sometimes the full global symmetry is ``hidden'' on the
magnetic side. It is not realized as a symmetry on the fundamental fields in
the Lagrangian. Rather, it emerges as a symmetry of the quantum theory. We then
focus on an approach based on self-dual models. We construct duals for some
very special matter content of $E_ 7$, $E_6$ and $F_4$. Again we find that the
full global symmetry is not realized on the fundamental fields.
