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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 (linkarticle, arXiv version).
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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 (link).
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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.