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