  In this expository paper, we explain a formula for the multiplicities of the
index of an equivariant transversally elliptic operator on a $G$-manifold. The
formula is a sum of integrals over blowups of the strata of the group action
and also involves eta invariants of associated elliptic operators. Among the
applications is an index formula for basic Dirac operators on Riemannian
foliations, a problem that was open for many years. This paper summarizes the
work in the papers arXiv:1005.3845 [math.DG] and arXiv:1008.1757 [math.DG].
