In this thesis we have introduced foliated rho-invariants on measured foliations and proved some of their stability properties. In particular, we have proved that the foliated rho-invariant associated with the leafwise signature operator on the foliation is independent of the leafwise metric on the foliation and is a leafwise diffeomorphism invariant, extending the classical result of Cheeger and Gromov for rho-invariants on coverings. In this work we have also obtained a generalization of Atiyahs Gamma-index theorem for foliations, which was known among experts but a proof has not appeared in the literature, as far as we know. We have also given an extension of the formalism of Hilbert-Poincaré (HP) complexes to the case of foliations and constructed an explicit homotopy equivalence of HP-complexes on leafwise homotopy equivalent foliations. This gives a direct proof of an already known result of the homotopy invariance of the signature index class for foliations. Finally, we have given an application of this formalism for partially extending the proof of the homotopy invariance of the classical Cheeger-Gromov rho-invariant to the foliated case.