  This is a review of old and new results and methods related to the Yau
conjecture on the zero set of Laplace eigenfunctions.
  The review accompanies two lectures given at the conference CDM 2018. We
discuss the works of Donnelly and Fefferman including their solution of the
conjecture in the case of real-analytic Riemannian manifolds.
  The review exposes the new results for Yau's conjecture in the smooth
setting. We try to avoid technical details and emphasize the main ideas of the
proof of Nadirashvili's conjecture. We also discuss two-dimensional methods to
study zero sets.
