N=1* SYM theories are conjectured to inherit the electromagnetic duality 
of N=4 Supersymmetric Yang-Mills.  This duality is non-trivially realized 
on the structure of massive vacua of the N=1* theory, owing to the modular 
properties of the theory.  These symmetry properties are sometimes alone 
enough to establish the exact superpotential of the theory.  We review the 
connection of N=1* with integrable systems, where in certain limits the 
exact solution is known.  For the gauge group SU(N), we review the approach 
of compactified theory on R^3 x S^1 where the exacts results can be obtained 
more explicitly.  Investigation of the case of non-simply-laced gauge groups, 
with more diverse vacuum structure, is work in progress.