Property (T) is a fundamental property of groups with applications in several areas of mathematics, including operator algebras and index theory. In this talk I will present a proof of property (T) for a new group, SAut(F_5), the special automorphism group of the free group on 5 generators. In particular, our method gives an explicit estimate of the Kazhdan constant of this group with respect to its natural set of generators, the Nielsen transformation. As a consequence the groups Aut(F_5) and Out(F_5) have property (T). I will also discuss applications to expanders graphs and related open problems. Joint work with Marek Kaluba and Narutaka Ozawa.