We analyse three different New Physics scenarios for Delta F=2 flavour-changing neutral currents in the quark sector in the light of recent data on neutral-meson mixing. We parametrise generic New Physics contributions to B_q-Bbar_q mixing (q=d,s), in terms of one complex quantity Delta_q, while three parameters Delta_K^tt, Delta_K^ct and Delta_K^cc are needed to describe K-Kbar mixing. In Scenario I, we consider uncorrelated New Physics contributions in the B_d, B_s, and K sectors. In this scenario, it is only possible to constrain the parameters Delta_d and Delta_s whereas there are no non-trivial constraints on the kaon parameters. In Scenario II, we study the case of Minimal Flavour Violation (MFV) and small bottom Yukawa coupling and Scenario III is the generic MFV case with large bottom Yukawa couplings. Our quantitative analyses consist of global CKM fits within the Rfit frequentist statistical approach, determining the Standard Model parameters and the new physics parameters of the studied scenarios simultaneously. We find that the recent measurements indicating discrepancies with the Standard Model are well accomodated in Scenarios I and III with new mixing phases, with a slight preference for Scenario I that permits different new CP phases in the B_d and B_s systems. Within our statistical framework, we find evidence of New Physics in both B_d and B_s systems. The Standard-Model hypothesis Delta_d=Delta_s=1 is disfavoured with p-values of 3.6 sigma and 3.3 sigma in Scenarios I and III, respectively. We also present an exhaustive list of numerical predictions in each scenario. In particular, we predict the CP phase in B_s -> J psi phi and the difference between the B_s and B_d semileptonic asymmetries, which will be both measured by the LHCb experiment.
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