  We present a comprehensive study of the spectral and transport properties in
the Anderson--Holstein model both in and out of equilibrium using the
functional renormalization group (FRG). We show how the previously established
machinery of Matsubara and Keldysh FRG can be extended to include the local
phonon mode. Based on the analysis of spectral properties in equilibrium we
identify different regimes depending on the strength of the electron--phonon
interaction and the frequency of the phonon mode. We supplement these
considerations with analytical results from the Kondo model. We also calculate
the non-linear differential conductance through the Anderson--Holstein quantum
dot and find clear signatures of the presence of the phonon mode.
