  We investigate the effect of local Coulomb correlations on electronic
transport through a variety of coupled quantum dot systems connected to Fermi
liquid leads. We use a newly developed functional renormalization group scheme
to compute the gate voltage dependence of the linear conductance, the
transmission phase, and the dot occupancies. A detailed derivation of the flow
equations for the dot level positions, the inter-dot hybridizations, and the
effective interaction is presented. For specific setups and parameter sets we
compare the results to existing accurate numerical renormalization group data.
This shows that our approach covers the essential physics and is quantitatively
correct up to fairly large Coulomb interactions while being much faster, very
flexible, and simple to implement. We then demonstrate the power of our method
to uncover interesting new physics. In several dots coupled in series the
combined effect of correlations and asymmetry leads to a vanishing of
transmission resonances. In contrast, for a parallel double-dot we find
parameter regimes in which the two-particle interaction generates additional
resonances.
