  We analyse the constraints and some of the phenomenological implications of a
class of two Higgs doublet models where there are flavour-changing neutral
currents (FCNC) at tree level but the potentially dangerous FCNC couplings are
suppressed by small entries of the CKM matrix $V$. This class of models have
the remarkable feature that, as a result of a discrete symmetry of the
Lagrangian, the FCNC couplings are entirely fixed in the quark sector by $V$
and the ratio $v_2 /v_1$ of the vevs of the neutral Higgs. The discrete
symmetry is extended to the leptonic sector, so that there are FCNC in the
leptonic sector with their flavour structure fixed by the leptonic mixing
matrix. We analyse a large number of processes, including decays mediated by
charged Higgs at tree level, processes involving FCNC at tree level, as well as
loop induced processes. We show that in this class of models one has new
physical scalars beyond the standard Higgs boson, with masses reachable at the
next round of experiments.
