We present several results on the size of the nodal (zero) set for solutions of partial differential equations of elliptic and parabolic type. In particular, we show a sharp upper bound for the $(n-1)$-dimensional Hausdorff measure of the nodal sets of the eigenfunctions of regular analytic elliptic problems in ${\mathbb R}^n$. We also show certain more recent results concerning semilinear equations (e.g. Navier-Stokes equations) and equations with non-analytic coefficients. The results on the size of nodal sets are connected to quantitative unique continuation, i.e., on the estimate of the order of vanishing of solutions of PDEs at a point. The results on unique continuation are joint with Ignatova and Camliyurt.