  The coming flavour precision era will allow to uncover various patterns of
flavour violation in different New Physics (NP) scenarios that are presently
washed out by hadronic and experimental uncertainties. We illustrate this by
performing the anatomy of flavour violation in the 331 models, based on the
gauge group SU(3)_C X SU(3)_L X U(1)_X that are among the simplest NP scenarios
with new sources of flavour and CP violation. The latter originate dominantly
through the flavour violating interactions of ordinary quarks and leptons with
a new heavy Z' gauge boson. After discussing first these models in some
generality, we present a detailed study of Delta F=2 observables and of rare K
and B meson decays in the specific beta=1/\sqrt{3} model (to be called
overline{331} model) assuming significantly smaller uncertainties in CKM and
hadronic parameters than presently available. The most prominent roles in our
analysis play epsilon_K, Delta M_q (q=d,s), the mixing induced CP asymmetries
S_{psi K_S} and S_{psi phi}, rare decays B_{s,d}-> mu^+ mu^- and in particular
the CP-asymmetry S_{mu^+ mu^-}^s in B_s ->mu^+ mu^-. We identify a number of
correlations between various observables that differ from those known from CMFV
models. While, the Delta F=2 observables allow still for four oases in the new
parameter space, we demonstrate how the inclusion of Delta F=1 observables can
in the future identify the optimal oasis for this model. For 1 TeV <= M_{Z'}<=
3 TeV the overline{331} model is in good agreement with the recent data for
B^+->tau^+ nu_tau, it removes the epsilon_K-S_{psi K_S} tension present in the
SM and the epsilon_K-Delta M_{s,d} tension present in CMFV models.
Simultaneously, while differing from the SM, it is consistent with the present
data on BR(B_{s,d}->mu^+ mu^-) and S_{psi phi}. We identify the triple
correlation BR(B_s->mu^+ mu^-)-S_{psi phi}-S_{mu^+ mu^-}^s.
