  We describe the Cartan and Weil models of twisted equivariant cohomology
together with the Cartan homomorphism among the two, and we extend the
Chern-Weil homomorphism to the twisted equivariant cohomology. We clarify that
in order to have a cohomology theory, the coefficients of the twisted
equivariant cohomology must be taken in the completed polynomial algebra over
the dual Lie algebra of $G$. We recall the relation between the equivariant
cohomology of exact Courant algebroids and the twisted equivariant cohomology,
and we show how to endow with a generalized complex structure the finite
dimensional approximations of the Borel construction $M\times_G EG_k$, whenever
the generalized complex manifold $M$ possesses a Hamiltonian $G$ action.
