  We present a detailed description of the three inequivalent twists of N=4
supersymmetric gauge theories. The resulting topological quantum field theories
are reobtained in the framework of the Mathai-Quillen formalism and the
corresponding moduli spaces are analyzed. We study their geometric features in
each case. In one of the twists we make contact with the theory of non-abelian
monopoles in the adjoint representation of the gauge group. In another twist we
obtain a topological quantum field theory which is orientation reversal
invariant. For this theory we show how the functional integral contributions to
the vacuum expectation values leading to topological invariants notably
simplify.
