In 1978, V. Ivrii conjectured that for a domain $\Omega\subset\mathbb{R}^n$ with sufficiently smooth boundary, the set of periodic orbits in the corresponding billiard has measure zero. He required this geometric assumption for his estimate of the asymptotics of the eigenvalues of the Laplacian operator. When he came with this conjecture to Sinai seminar, the members of the seminar said that they would solve this problem in a couple of weeks. More than 35 years later, the problem is still open. Ivrii’s conjecture is proved for a generic domain with smooth boundary and for triangular orbits in any (not necessarily generic) domain. A few years ago, Alexey Glutsyuk and I proved that for a domain $\Omega\subset\mathbb{R}^2$ with piecewise $\mathcal{C}^4$ smooth boundary, the set of quadrilateral periodic orbits has measure zero. In the talk I will highlight the main ideas of the proof.