  Using adiabatic limits of Eta invariants, Rho invariants of the total space
of a fiber bundle are investigated. One concern is to formulate the aspects of
local index theory for families of Dirac operator in terms of the odd signature
operator, and place known results in a context which permits the treatment of
Rho invariants. The main focus is, however, to use this to compute Rho
invariants for explicit classes of fibered 3-manifolds. More precisely, we
consider $\U(1)$-Rho invariants of principal $S^1$-bundles over closed,
oriented surfaces as well as mapping tori with torus fiber. Hyperbolic
monodromy maps deserve particular attention. When discussing them, the
logarithm of a generalized Dedekind Eta function naturally appears. An explicit
formula for $\U(1)$-Rho invariants is then deduced from a transformation
formula for these Dedekind Eta functions.
