We will first discuss the classical Tits alternative about finitely generated linear groups. Then we will present a strong refinement of it, which has several consequences about the growth, the number of relations in a given presentation, as well as spectral properties of the regular representation of a finitely generated linear group. We will also give applications to girth and expansion properties of finite subgroups of GL(d, Fp). If time permits we hope to present some of the number theoretical ideas, involving diophantine geometry, which underlie the proofs.