If r is a symmetric Lie subalgebra (i.e. fixed under an involution) of a semisimple Lie algebra g and V is a g-module then Vogan has introduced the notion of Dirac cohomology HD(V ). He has also made conjectures about the action of Z(g) (the center of U(g)) on HD(V ). These conjectures have been proved by Huang and Pandzic in a paper to appear in JAMS. Using the the cubic Dirac operator, which we have introduced for other purposes, we extend the notion of Dirac cohomology and show the results of Huang and Pandzic extend to the case where r is any reductive Lie subalgebra of g so long as B|r is non-singular. Here B is a symmetric ad-invariant bilinear form on g. Involved is the determination of a homomorphism η : Z(g) → Z(r). The determination of η depends upon showing the existence of sufficiently many modules V for which HD(V ) 6= 0. The homomorphism η is such that, using a famous result of H.Cartan, H ∗ (G/R, C) = TorZ( ) (C, Z(