Cubic Curves and Cubic Surfaces and Cubic Threefolds and Cubic Fourfolds

Rationality questions are among the oldest, most natural, and most stubborn problems in algebraic geometry. The simplest varieties whose rationality is interesting are the cubic hypersurfaces, and already the matter is deep and full of surprises. I will explain the well-known classical answers for cubic curves and surfaces, sketch the proof of the landmark Clemens—Griffiths result for cubic threefolds, and attempt to say something reasonably coherent about the recent breakthrough by Katzarkov—Kontsevich—Pantev—Yu in dimension four.