We present a general analysis of extensions of the Standard Model which satisfy the criterion of Minimal Flavour Violation (MFV). We define this general framework by constructing a low-energy effective theory containing the Standard Model fields, with one or two Higgs doublets and, as the only source of SU(3) 5 flavour symmetry breaking, the background values of fields transforming under the flavour group as the ordinary Yukawa couplings. We analyse present bounds on the effective scale of dimension-six operators, which range between 1 and 10 TeV, with the most stringent constraints imposed by B→ Xsγ. In this class of theories, it is possible to relate predictions for FCNC processes in B physics to those in K physics. We compare the sensitivity of various experimental searches in probing the hypothesis of MFV. Within the two-Higgs-doublet scenario, we develop a general procedure to obtain all tan β-enhanced Higgs-mediated FCNC amplitudes, discussing in particular their impact in B→ℓ +ℓ -, ∆MB and B→ Xsγ. As a byproduct, we derive some two-loop tan β-enhanced supersymmetric contributions to B→ Xsγ previously unknown.