  We determine the exact global structure of the moduli space of $N{=}2$
supersymmetric $SO(n)$ and $\USp(2n)$ gauge theories with matter
hypermultiplets in the fundamental representations, using the
non-renormalization theorem for the Higgs branches and the exact solutions for
the Coulomb branches. By adding an $(N{=}2)$--breaking mass term for the
adjoint chiral field and varying the mass, the $N{=}2$ theories can be made to
flow to either an ``electric'' $N{=}1$ supersymmetric QCD or its $N{=}1$ dual
``magnetic'' version. We thus obtain a derivation of the $N{=}1$ dualities of
Seiberg.
