In recent years several experiments revealed intriguing hints for new physics (NP) in $B$ meson decays involving \bctaunu and $b\to s\ell^+\ell^-$ transitions at the $4-5\,\sigma$ level. These signs for NP are supported by slight disagreements in the analogous observables with $b\to u \tau\nu$ and $b\to d\mu^+\mu^-$ transitions. While not significant on their own, these processes point in the same direction.
Since all these observables are semi-leptonic, leptoquarks are prime candidates for a solution to these puzzels since they give tree-level effects here while their contributions to other flavour observables, which agree with the SM predictions, are loop suppressed.
In these proceedings we focus on the vector leptoquark $SU(2)$ singlet which provides a natural explanations for both the $b\to c(u) \tau\nu$ and the $b\to s(d)\mu^+\mu^-$ data. However, to account for the preferred NP effect in $b\to c(u) \tau\nu$ large couplings to $\tau$ leptons are necessary and loop effects become important. Including them in the phenomenological analysis, the fit to data is even improved.
Finally, we review two UV complete models which give rise to the desired vector leptoquark $SU(2)$ singlet. The first possibility is to add either three generations of vector-like fermions to the original Pati-Salam model and break the symmetry with a Higgs mechanism. Alternatively, it is possible to break the PS gauge group to the SM one on a compact extra dimension like in Randall Sundrum models.