We shall present the circle of ideas concerning the coarse geometric approach to the Baum-Connes conjecture, via the analytic surgery sequence of N. Higson and J. Roe. In joint work with M.-T. Benameur, we elicit its relations with some secondary invariants of Dirac operators on co-compact coverings, which generalize classical deep results originally due to N. Keswani. We have now partially extended this program further to the context of non-compact, complete spin foliations, which provide new obstructions to the existence of leafwise metrics of (uniformly) positive scalar curvature via the coarse index, generalizing the results of Connes for smooth foliations of compact manifolds. If time permits, we shall also outline the connections of this framework with the work of Gromov-Lawson.