  We show that the strategy of point counting in o-minimal structures can be
applied to various problems on unlikely intersections that go beyond the
conjectures of Manin-Mumford and Andr\'e-Oort. We verify the so-called
Zilber-Pink Conjecture in a product of modular curves on assuming a lower bound
for Galois orbits and a sufficiently strong modular Ax-Schanuel Conjecture. In
the context of abelian varieties we obtain the Zilber-Pink Conjecture for
curves unconditionally when everything is defined over a number field. For
higher dimensional subvarieties of abelian varieties we obtain some weaker
results and some conditional results.
