  We explicitly determine the global structure of the $SL(2,Z)$ bundle over the
Coulomb branch of the moduli space of asymptotically free $N=2$ supersymmetric
Yang-Mills theories with gauge group $SU(2)$ when massless hypermultiplets are
present. For each relevant number of flavours, we show that there is a curve of
marginal stability on the Coulomb branch, diffeomorphic to a circle, across
which the BPS spectrum is discontinuous. We determine rigorously and completely
the BPS spectra inside and outside the curve. In all cases, the spectrum inside
the curve consists of only those BPS states that are responsible for the
singularities of the low energy effective action (in addition to the massless
abelian gauge multiplet which is always present). The predicted decay patterns
across the curve of marginal stability are perfectly consistent with all
quantum numbers carried by the BPS states. As a byproduct, we also show that
the electric and magnetic quantum numbers of the massless states at the
singularities proposed by Seiberg and Witten are the only possible ones.
