It is possible that some nuclei may attain even greater quadrupole deformations than those observed in superdeformed nuclei and the search for such hyperdeformed shapes has been a goal of nuclear structure for many years. In this talk I will discuss recent developments in the experimental study of superdeformed nuclei and in particular the search for the more exotic hyperdeformed nuclei. I will begin by briefly reviewing the basic concepts connected with nuclei at extreme deformations and introduce the use of intruder occupancy as a method to classify the nuclear shape (normal, super, or hyperdeformed) before turning to our recent attempts to extend these studies from super to hyperdeformations; the focus will be on our work in the cadmium and tin region. Finally, I will say a few words on future prospects.

PACS numbers: 21.10.Re, 23.20.Lv, 29.30.Kv