  The asymptotic dimension of metric spaces is an important notion in geometric
group theory introduced by Gromov. The metric spaces considered in this paper
are the ones whose underlying spaces are the vertex-sets of graphs and whose
metrics are the distance functions in graphs. A standard compactness argument
shows that it suffices to consider the asymptotic dimension of classes of
finite graphs.
  In this paper we prove that the asymptotic dimension of any proper
minor-closed family, any class of graphs of bounded tree-width, and any class
of graphs of bounded layered tree-width are at most 2, 1, and 2, respectively.
The first result solves a question of Fujiwara and Papasoglu; the second and
third results solve a number of questions of Bonamy, Bousquet, Esperet,
Groenland, Pirot and Scott. These bounds for asymptotic dimension are optimal
and improve a number of results in the literature. Our proofs can be
transformed into linear or quadratic time algorithms for finding coverings
witnessing the asymptotic dimension which is equivalent to finding weak
diameter colorings for graphs. The key ingredient of our proof is a unified
machinery about the asymptotic dimension of classes of graphs that have
tree-decompositions of bounded adhesion over hereditary classes with known
asymptotic dimension, which might be of independent interest.
