An introduction to various Diophantine problems in number theory, with approaches involving analytic methods. The exact subject matter of the course will depend on the participants and general interest, potential topics could include the basic theory of Abelian Varieties, the Lang conjecture, the Manin-Mumford and Andre-Oort conjectures, as well as the Bogomolov conjecture and the equidistribution and large Galois orbits.