  We discuss a variation of Gromov's notion of asymptotic dimension that was
introduced and named Nagata dimension by Assouad. The Nagata dimension turns
out to be a quasisymmetry invariant of metric spaces. The class of metric
spaces with finite Nagata dimension includes in particular all doubling spaces,
metric trees, euclidean buildings, and homogeneous or pinched negatively curved
Hadamard manifolds. Among others, we prove a quasisymmetric embedding theorem
for spaces with finite Nagata dimension in the spirit of theorems of Assouad
and Dranishnikov, and we characterize absolute Lipschitz retracts of finite
Nagata dimension.
