  We derive the exact effective superpotential in 4d, N=1 supersymmetric SU(2)
gauge theories with $N_A$ triplets and $2N_f$ doublets of matter superfields.
We find the quantum vacua of these theories; the equations of motion (for
$N_A=1$) can be reorganized into the singularity conditions of an elliptic
curve. From the phase transition points to the Coulomb branch, we find the
exact Abelian gauge couplings, $\tau$, for arbitrary bare masses and Yukawa
couplings. We thus {\em derive} the result that $\tau$ is a section of an
$SL(2,\Z)$ bundle over the moduli space and over the parameters space of bare
masses and Yukawa couplings. For $N_c>2$, we derive the exact effective
superpotential in branches of supersymmetric $SU(N_c)$ gauge theories with one
supermultiplet in the adjoint representation ($N_A=1$) and zero or one flavor
($N_f=0,1$). We find the quantum vacua of these theories; the equations of
motion can be reorganized into the singularity conditions of a genus $N_c-1$
hyperelliptic curve. Finally, we present the effective superpotential in the
$N_A$, $N_f<N_c$ cases.
