  We compare the effectiveness of two hypotheses, Natural Flavour Conservation
(NFC) and Minimal Flavour Violation (MFV), in suppressing the strength of
flavour-changing neutral-currents (FCNCs) in models with more than one Higgs
doublet. We show that the MFV hypothesis, in its general formulation, is more
stable in suppressing FCNCs than the hypothesis of NFC alone when quantum
corrections are taken into account. The phenomenological implications of the
two scenarios are discussed analysing meson-antimeson mixing observables and
the rare decays B -> mu+ mu-. We demonstrate that, introducing flavour-blind CP
phases, two-Higgs doublet models respecting the MFV hypothesis can accommodate
a large CP-violating phase in Bs mixing, as hinted by CDF and D0 data and,
without extra free parameters, soften significantly in a correlated manner the
observed anomaly in the relation between epsilon_K and S_psi_K.
