  Tests of lepton-universality as rate ratios in $b\to s \ell\ell$ transitions
can be predicted very accurately in the Standard Model. The deficits with
respect to expectations reported by the LHCb experiment in muon-to-electron
ratios of the $B\to K^{(*)}\ell\ell$ decay rates thus point to genuine
manifestations of lepton non-universal new physics. In this paper, we analyse
these measurements in the context of effective field theory. First, we discuss
the interplay of the different operators in $R_K$ and $R_{K^*}$ and provide
predictions for $R_{K^*}$ in the Standard Model and in new-physics scenarios
that can explain $R_K$. We also provide approximate numerical formulas for
these observables in bins of interest as functions of the relevant Wilson
coefficients. Secondly, we perform frequentist fits to $R_{K}$ and $R_{K^*}$.
The SM disagrees with these measurements at $3.7\sigma$ significance. We find
excellent fits in scenarios with combinations of $\mathcal O_{9(10)}^\ell=\bar
s\gamma^\mu b_L~\ell\gamma_\mu(\gamma_5) \ell$ operators, with pulls relative
to the Standard Model in the region of $4\sigma$. An important conclusion of
our analysis is that a lepton-specific contribution to $\mathcal O_{10}$ is
essential to understand the data. Under the hypothesis that new-physics couples
selectively to the muons, we also present fits to other $b\to s\mu\mu$ data
with a conservative error assessment, and comment on more general scenarios.
Finally, we discuss new lepton universality ratios that, if new physics is the
origin of the observed discrepancy, should contribute to the statistically
significant discovery of new physics in the near future.
