We generalize the notions of asymptotic dimension and coarse embeddings from metric spaces to quantum metric spaces in the sense of Kuperberg and Weaver. We show that quantum asymptotic dimension behaves well with respect to metric quotients, direct sums, and quantum coarse embeddings. Moreover, we prove that a quantum metric space that equi-coarsely contains a sequence of reflexive quantum expanders must have infinite asymptotic dimension. This is done by proving a vertex-isoperimetric inequality for quantum expanders, based upon a previously known edge-isoperimetric one due to Temme, Kastoryano, Ruskai, Wolf, and Verstraete. Joint work with Andrew Swift.