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Abstract: Arguably, the main goal of algebraic geometry is to classify algebraic varieties, up to birational equivalence. Even the simplest case, determining whether a variety is birational to projective space, is incredibly subtle. We will discuss this question in the context of cubic hypersurfaces, reviewing the current state of the art rationality conjectures. We will discuss recent progress in understanding the birational geometry of cubic fourfolds using new tools such as the derived category, equivariant geometry, and the geometry of hyperkähler manifolds.