The presence of approximate flavour symmetries is invoked to make New Physics models near the TeV scale compatible with the very demanding quark flavour violation tests. These tests pose a serious threat for any acceptable Standard Model extension at the TeV scale, motivated by naturalness issues. Suitable approximate flavour symmetries can provide an appropriate suppression to unwanted flavour effects, while leading to peculiar patterns of small, but potentially detectable in forthcoming experiments, deviations in flavour observables. After reviewing the two most popular options in the literature for such symmetries (U(2)3 and U(3)3), we focus on lepton flavour violation (LFV) which also gives very severe experimental bounds. Making U(2)3 and U(3)3 compatible with SU(5) Grand Unified flavour group allows us to make quantitative predictions about LFV processes (...). Using an Effective Field Theory approach new effects in the quark sector are discussed as well and the explicit realization of this scenario in the specific case of Supersymmetry is analyzed in detail. Finally, we try to explain the recent tensions in B physics, hinting toward New Physics deviations of lepton flavour universality in semileptonic decays, in the context of a U(2)5 flavour symmetry and a tree level leptoquark (LQ) exchange. The current experimental deviations can be accommodated invoking a vector-electroweak singlet LQ with a relatively large value (below 1 TeV) of the ratio between its mass and its dominant coupling. A partial UV completion of this scenario based on a SU(4) x SO(5) x U(1)X Composite Higgs model with a U(2)5 flavour symmetry is examined. Special attention is given to new effects in both indirect (flavour and electroweak) observables and collider signatures. The original part of this thesis is contained in chapters 4, 5, 6 and it is mainly based on the work published in the papers [1, 2, 3].