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.