  We analyse the interplay of the constraints imposed on flavour-symmetric
Composite Higgs models by Naturalness considerations and the constraints
derived from Flavour Physics and Electroweak Precision Tests. Our analysis is
based on the Effective Field Theory which describes the Higgs as a
pseudo-Nambu-Goldstone boson and also includes the composite fermionic
resonances. Within this approach one is able to identify the directions in the
parameter space where the $U(3)$-symmetric flavour models can pass the current
experimental constraints, without conflicting with the light Higgs mass. We
also derive the general features of the $U(2)$-symmetric models required by the
experimental bounds, in case of elementary and totally composite $t_R$. An
effect in the $Z \bar b b$ coupling, which can potentially allow for sizable
deviations in $Z \to \bar b b$ decay parameters without modifying flavour
physics observables, is identified. We also present an analysis of the mixed
scenario, where the top quark mass is generated due to Partial Compositeness
while the light quark masses are Technicolor-like.
