  In view of the emerging hints for the violation of lepton flavour
universality in several $B$-meson decays, we conduct a model-independent study
(effective field theory approach) of several well-motivated new physics
scenarios. Taking into account the most recent LHCb data, we provide updates to
Wilson coefficient fits for numerous popular new physics hypotheses. We also
consider a promising model of vector leptoquarks, which in addition to
explaining the $B$-meson decay anomalies ($R_{K^{(*)}}$ and $R_{D^{(*)}}$)
would have an extensive impact for numerous flavour observables. We identify
promising decay modes allowing to (indirectly) probe such an extension: these
include positive signals (at Belle II or LHCb) for $\tau\to \phi \mu$,
$B_{(s)}$-meson decays to $\tau^+ \tau^-$ and $\tau^+ \mu^-$ ($\tau^+ e^-$)
final states, as well as an observation of certain charged lepton flavour
violation transitions at COMET and Mu2e. We also argue how the evolution of the
experimental determination of $R_{D^{(*)}}$ can prove instrumental in
falsifying a vector leptoquark explanation of the anomalous $B$-meson decay
data.
