In this talk, I will address the following question, attributed to Gizatullin: Which automorphisms of a smooth quartic surface in projective 3-space are restrictions of Cremona transformations of the ambient space? Techniques from birational geometry, and in particular the Minimal Model Program, provide powerful tools for studying this problem. I will report on recent progress on Gizatullin's problem, obtained in joint works with Alessio Corti and Alex Massarenti, and with Daniela Paiva and Sokratis Zikas.