  This thesis presents a thorough analysis of the links between the N=4
supersymmetric gauge theory in four dimensions and its three topological
twisted counterparts. Special emphasis is put in deriving explicit results in
terms of the vacuum structure and low-energy effective description of
four-dimensional supersymmetric gauge theories. A key ingredient is the
realization of Montonten-Olive duality in the twisted theories, which is
discussed in detail from different viewpoints.
