  In 2015, Felsner, Trotter, and Wiechert showed that posets with outerplanar
cover graphs have bounded dimension. We generalise this result to posets with
$k$-outerplanar cover graphs. Namely, we show that posets with $k$-outerplanar
cover graph have dimension $\mathcal{O}(k^3)$. As a consequence, we show that
every poset with a planar cover graph and height $h$ has dimension
$\mathcal{O}(h^3)$. This improves the previously best known bound of
$\mathcal{O}(h^6)$ by Kozik, Micek and Trotter.
