In this talk we will study the stability properties of rho-invariants (also known as relative eta-invariants) which were introduced by Atiyah, Patodi and Singer for closed manifolds, in the case of measured foliations. In particular we will show that the "foliated" rho-invariant associated with a leafwise signature operator is independent of the leafwise metric on the foliation, extending the classical result of Cheeger and Gromov in the $L^2$ case. We will also outline the proof of the homotopy invariance of such invariants (this is still work in progress). In this context we will give an analogue of Atiyah's $L^2$-index theorem and also use a generalization of the machinery of Hilbert-Poincaré complexes of Higson and Roe, thereby obtaining a new proof of the leafwise homotopy invariance of the index of leafwise signature operators for foliations.