© 2017 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the «https://creativecommons.org/licenses/by/4.0/» Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Tests of lepton-universality as rate ratios in b→sℓℓ 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→K(∗)ℓℓ decay rates thus point to genuine manifestations of lepton nonuniversal new physics. In this paper, we analyze these measurements in the context of effective field theory. First, we discuss the interplay of the different operators in RK and RK∗ and provide predictions for RK∗ in the Standard Model and in new-physics scenarios that can explain RK. 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 RK and RK∗. The Standard Model disagrees with these measurements at 3.7σ significance. We find excellent fits in scenarios with combinations of O9(10)ℓ=s γμbLℓγμ(γ5)ℓ operators, with pulls relative to the Standard Model in the region of 4σ. An important conclusion of our analysis is that a lepton-specific contribution to O10 is important to understand the data. Under the hypothesis that new-physics couples selectively to the muons, we also present fits to other b→sμμ 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.