Birkhoff conjecture about integrable billiards 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.
 A: For a recent progress see http://arxiv.org/pdf/1412.2853.pdf
a local version of this conjecture is proven. 
A: I haven't heard of any recent breakthroughs. The strongest result that I know is due to  Misha Bialy:

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.

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, arXiv version).
A: I'm no expert, but according to Tabachnikov the conjecture was still open as of 2005, while Delshams and Ramirez-Ros have a local result (i.e. the conjecture is true when considering symmetric entire perturbations). Probably Mathscinet would help more.
A: I think has recently obtained partial results on this conjecture (On polynomially integrable Birkhoff billiards on surfaces of constant curvature, JEMS 2021)
