Cubic hypersurfaces of dimension three have played a central role in the development of Hodge theory. For example, the question whether cubic threefolds are rational has been answered by Clemens and Griffiths by means of intermediate Jacobians. The analogous question for cubic hypersurfaces of dimension four is still open but Hodge theory is again expected to play a key role. Instead of intermediate Jacobians, now K3 surfaces and certain categorical versions of them (also called Kuznetsov components) are used to detect (ir)rationality. In this talk I will survey our current understanding and compare various conjectural descriptions of rational cubic fourfolds.