This talk will survey two topics developed in joint work with Alain Connes, in which cyclic cohomology played a key role. I will first recall the local index formula for spectral triples with discrete $\zeta$-dimension spectrum, which represents an extension of the Atiyah-Singer-Bott-Patodi local index theorem to noncommutative spaces. I will then outline its application to the geometry of leaf-spaces of foliations, which in turn led us to devise the Hopf-cyclic cohomology. Lastly, I will illustrate how the latter forms a universal receptacle for the characteristic classes of foliations and potentially of other noncommutative spaces.