  A maximally supersymmetric configuration of super Yang-Mills living on a
noncommutative torus corresponds to a constant curvature connection. On a
noncommutative toroidal orbifold there is an additional constraint that the
connection be equivariant. We study moduli spaces of (equivariant) constant
curvature connections on noncommutative even-dimensional tori and on toroidal
orbifolds. As an illustration we work out the cases of Z_{2} and Z_{4}
orbifolds in detail. The results we obtain agree with a commutative picture
describing systems of branes wrapped on cycles of the torus and branes stuck at
exceptional orbifold points.
