  We show that planar graphs have bounded queue-number, thus proving a
conjecture of Heath, Leighton and Rosenberg from 1992. The key to the proof is
a new structural tool called layered partitions, and the result that every
planar graph has a vertex-partition and a layering, such that each part has a
bounded number of vertices in each layer, and the quotient graph has bounded
treewidth. This result generalises for graphs of bounded Euler genus. Moreover,
we prove that every graph in a minor-closed class has such a layered partition
if and only if the class excludes some apex graph. Building on this work and
using the graph minor structure theorem, we prove that every proper
minor-closed class of graphs has bounded queue-number.
  Layered partitions have strong connections to other topics, including the
following two examples. First, they can be interpreted in terms of strong
products. We show that every planar graph is a subgraph of the strong product
of a path with some graph of bounded treewidth. Similar statements hold for all
proper minor-closed classes. Second, we give a simple proof of the result by
DeVos et al. (2004) that graphs in a proper minor-closed class have low
treewidth colourings.
