There is a conjecture by Birkhoff which claims that for a simple closed $C^2$ plane curve $C$, if the billiard ball map is integrable then the curve is an ellipse. 


Integrability here might be formulated as follows: there exists a neighbourhood of $C$ in the interior $Int(C)$ that is foliated by caustics (caustics being curves that are everywhere tangent to a given trajectory of the billiard ball). 


I would be interested to know the current status (and progresses, if there are) of this conjecture.



Birkhoff conjecture about integrable billiards
I haven't heard of any recent breakthroughs. The strongest result that I know is due to [Misha Bialy](https://doi.org/10.1007/BF02572397):



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> **Theorem.** If almost every phase point of the billiard ball map in a strictly convex billiard table belongs to an invariant circle, then the billiard table is a disc.
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Stronger results are available for an outer version of the Birkhoff conjecture. Tabachnikov
proved that if the outer billiard map around a plane oval is algebraically integrable then the oval is an ellipse ([article](http://pjm.math.berkeley.edu/pjm/2008/235-1/pjm-v235-n1-p07-s.pdf), [arXiv version](https://arxiv.org/abs/0708.0255)).


