  The crossing number of a graph is the minimum number of crossings in a
drawing of the graph in the plane. Our main result is that every graph $G$ that
does not contain a fixed graph as a minor has crossing number $O(\Delta n)$,
where $G$ has $n$ vertices and maximum degree $\Delta$. This dependence on $n$
and $\Delta$ is best possible. This result answers an open question of Wood and
Telle [New York J. Mathematics, 2007], who proved the best previous bound of
$O(\Delta^2 n)$. We also study the convex and rectilinear crossing numbers, and
prove an $O(\Delta n)$ bound for the convex crossing number of bounded
pathwidth graphs, and a $\sum_v\deg(v)^2$ bound for the rectilinear crossing
number of $K_{3,3}$-minor-free graphs.
