A family of problems in Diophantine geometry has the following
form: We fix a collection of "special" algebraic varieties among which the
0-dimensional are called "special points". Mostly, if V is a special variety
then the special points are Zariski dense in V, and the problem is to prove
the converse: If V is an irreducible algebraic variety and the special
points are Zariski dense in V then V itself is special.
Particular cases of the above are the Manin-Mumford conjecture
(where the special varieties are certain cosets of abelian sub-varieties and
the special points are torsion points), the Mordell-Lang conjecture, the Andre-Oort Conjecture. In 1990's Hrushovski showed how methods of model theory could be applied to solve certain such problems. About 10 years ago Pila and Zannier developed a different framework which allows to apply model theory and especially the theory of o-minimal structures, to tackle questions of this nature over the complex numbers. Pila used these methods to prove some open cases of the Andre-Oort conjecture and since then there was an influx of articles which use similar techniques. At the heart
of the Pila-Zannier method lies a theorem of Pila and Wilkie on rational
points on so-called definable sets in o-minimal structures.
In these survey talks I will describe the basic mode theoretic ingredients of
the Pila-Zannier method and the way in which it is applied. I
will not assume prior knowledge of model theory or Logic and will try to
explain all notions which may come up.