Tits alternative for the Cremona group

A group G satisfies the Tits alternative if any subgroup contains either a free group over 2 generators, of a solvable group of finite index.
The classical result by Jacques Tits is that the linear groups satisfy this alternative, up to restricting to finitely generated subgroups in the case of a ground field of positive characteristic.  
In this talk I will explain the strategy of proof to show that the plane Cremona group also satisfies the Tits alternative.
It will essentially be a survey of previous results by Cantat and Urech, with a few contributions by myself mainly for the case of positive characteristic, and also for the classification of torsion subgroups.