We determine the exact global structure of the moduli space of N = 2 s
upersymmetric SO(n) and USp(2n) gauge theories with matter hypermultip
lets in the fundamental representations, using the non-renormalization
theorem for the Higgs branches and the exact solutions for the Coulom
b branches. By adding an (N = 2)-breaking mass term for the adjoint ch
iral field and varying the mass, the N = 2 theories can be made to flo
w to either an ''electric'' N = 1 supersymmetric QCD or its N = 1 dual
''magnetic'' version. We thus obtain a derivation of the N = 1 dualit
ies of Seiberg.