The recent measurement of RK∗ is yet another hint of new physics (NP) and supports the idea that it is present in b → sμμ- decays. We perform a combined model-independent and model-dependent analysis in order to deduce properties of this NP. Like others, we find that the NP must obey one of two scenarios: (I) Cμμ 9 NP < 0 or (II) Cμμ 9 NP = -Cμμ 10 NP < 0. A third scenario, (III) Cμμ 9 NP = -C0μμ 9 NP, is rejected largely because it predicts RK = 1, in disagreement with experiment. The simplest NP models involve the tree-level exchange of a leptoquark (LQ) or a Z0 boson.We show that scenario II can arise in LQ or Z0 models, but scenario I is only possible with a Z0. Fits to Z0 models must take into account the additional constraints from B0s - B0s mixing and neutrino trident production. Although the LQs must be heavy, O(TeV), we find that the Z0 can be light, e.g., MZ0 = 10 GeV or 200 MeV. © 2017 American Physical Society.