**In minisymposium: Recent Developments in Non-linear Preservers**

In many preserver problems bijectivity of the preserver is not an assumption in the theorem that characterizes it, but a conclusion. Often the complexity of the proof of such theorem reduces significantly if one assumes bijectivity right from the beginning. In the talk I will present some techniques and results on certain preserver problems, where tools from graph theory can be used to prove that bijectivity of the preserver is obtained automatically. Clearly, these tools do not work if there exist non-bijective preservers. It is also true that sometimes these tools fail, though all preservers turn out to be bijective. An increasing number of examples indicate that in this last situation the preserver problem can be related, or even equivalent, to certain (open and well known) problems in finite geometry.