In this talk I will present joint work with Andreas Thom on bounded normal generation (BNG) for projective unitary groups of von Neumann algebras. We say that a group has (BNG) if the conjugacy class of every nontrivial element and of its inverse generate the whole group in finitely many steps. After explaining how one can prove (BNG) for the projective unitary group of a finite factor, I will present applications to automatic continuity of homomorphisms with SIN target groups. The talk will be closed with recent results on the Bergman property for unitary groups of II_1 factors and countable cofinality for compact connected Lie groups.