We show that for every integer n⩾1$n\geqslant 1$, there exists a graph Gn$G_n$ with (1+o(1))n$(1+o(1))n$ vertices and n1+o(1)$n^{1 + o(1)}$ edges such that every n$n$‐vertex planar graph is isomorphic to a subgraph of Gn$G_n$. The best previous bound on the number of edges was O(n3/2)$O(n^{3/2})$, proved by Babai, Chung, Erdős, Graham, and Spencer in 1982. We then show that for every integer n⩾1$n\geqslant 1$, there is a graph Un$U_n$ with n1+o(1)$n^{1 + o(1)}$ vertices and edges that contains induced copies of every n$n$‐vertex planar graph. This significantly reduces the number of edges in a recent construction of the authors with Dujmović, Gavoille, and Micek.