Birkhoff's conjecture on integrable billiards

Let $\Omega\subset\mathbb{R}^2$ be a bounded convex domain whose boundary $\partial\Omega$ is a $C^2$-smooth convex curve. Suppose that a neighborhood of $\partial\Omega$ is foliated by caustics. Birkhoff's conjecture. Then $\partial\Omega$ is an ellipse. Although this conjecture remains open, substantial work has established results under additional hypotheses and developed partial approaches to the characterization of elliptic billiards.