  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 ->
X_s gamma. 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(beta)-enhanced Higgs-mediated FCNC amplitudes, discussing in particular
their impact in B -> l^+l^-, Delta M_B and B -> X_s gamma. As a byproduct, we
derive some two-loop tan(beta)-enhanced supersymmetric contributions to B ->
X_s gamma previously unknown.
