$B$ Flavour Anomalies: 2021 Theoretical Status Report

At the present time, there are discrepancies with the predictions of the SM in several observables involving $b \to s \ell^+ \ell^-$ and $b \to c \ell^- {\bar\nu}_\ell$ decays. These are the $B$ flavour anomalies. In this review, we summarize the data as of Moriond 2021 and present theoretical new-physics explanations from both a model-independent effective-field-theory point of view and through the building of explicit models. Throughout, we stress the complementarity of these two approaches. We also discuss combined explanations of both $B$ anomalies, and present models that also explain other problems, such as dark matter, $(g-2)_\mu$, neutrino properties, and hadronic anomalies.