  We show that there exists an adjacency labelling scheme for planar graphs
where each vertex of an $n$-vertex planar graph $G$ is assigned a
$(1+o(1))\log_2 n$-bit label and the labels of two vertices $u$ and $v$ are
sufficient to determine if $uv$ is an edge of $G$. This is optimal up to the
lower order term and is the first such asymptotically optimal result. An
alternative, but equivalent, interpretation of this result is that, for every
$n$, there exists a graph $U_n$ with $n^{1+o(1)}$ vertices such that every
$n$-vertex planar graph is an induced subgraph of $U_n$. These results
generalize to bounded genus graphs, apex-minor-free graphs, bounded-degree
graphs from minor closed families, and $k$-planar graphs.
