We will discuss the recent work of Logunov and Malinnikova, for which they were awarded a Clay research award. This work led to the proof of Nadirashvili's conjecture and of the lower bound in Yau's conjecture. Nadirashvili's conjecture roughly states that if $u$ is a harmonic function in the unit ball in $\mathbb{R}^n$ which is zero at the center of the ball, then the ($n-1$)-dimensional Hausdorff measure of the zero set of $u$ in the ball is bounded below by a purely dimensional constant. Yau's conjecture concerns the ($n-1$)-dimensional Hausdorff measure of the zero set of Laplace eigenfunctions. In particular, Yau conjectured that the zero set of the eigenfunction associated to the eigenvalue $\lambda$ should have ($n-1$)-dimensional Hausdorff measure comparable to $\sqrt{\lambda}$.