  We study systematically the BPS spectra of N=2 SYM coupled to
half--hypermultiplets, the basic example being E_7 SYM coupled to a half--hyper
in the 56 irrepr. In order to do this, we determine the BPS quivers with
superpotential of such N=2 models using a new technique we introduce. The
computation of the BPS spectra in the various chambers is then reduced to the
Representation Theory of the resulting quivers. We use the quiver description
to study the BPS spectrum at both strong and weak coupling. The following
models are discussed in detail: SU(6) SYM coupled to a half 20, SO(12) SYM
coupled to a half 32, and E_7 SYM coupled to a half 56. For models with gauge
group SU(2) x SO(2n) and matter in the half (2,2n) we find strongly coupled
chambers with a BPS spectrum consisting of just finitely many hypermultiplets.
