  Flavour- and CP-violating electromagnetic or chromomagnetic dipole operators
in the quark sector are generated in a large class of new physics models and
are strongly constrained by measurements of the neutron electric dipole moment
and observables sensitive to flavour-changing neutral currents, such as the
$B\to X_s\gamma$ branching ratio and $\epsilon'/\epsilon$. After a
model-independent discussion of the relevant constraints, we analyze these
effects in models with partial compositeness, where the quarks get their masses
by mixing with vector-like composite fermions. These scenarios can be seen as
the low-energy limit of composite Higgs or warped extra dimensional models. We
study different choices for the electroweak representations of the composite
fermions motivated by electroweak precision tests as well as different flavour
structures, including flavour anarchy and $U(3)^3$ or $U(2)^3$ flavour
symmetries in the strong sector. In models with "wrong-chirality" Yukawa
couplings, we find a strong bound from the neutron electric dipole moment,
irrespective of the flavour structure. In the case of flavour anarchy, we also
find strong bounds from flavour-violating dipoles, while these constraints are
mild in the flavour-symmetric models.
