  We provide a complete classification, up to order-isomorphism, of all
possible Wadge hierarchies on zero-dimensional Polish spaces using
(essentially) countable ordinals as complete invariants. We also observe that
although our assignment of invariants is very simple and there are only
$\aleph_1$-many equivalence classes, the above classification problem is quite
complex from the descriptive set-theoretic point of view: in particular, there
is no Borel procedure to determine whether two zero-dimensional Polish spaces
have isomorphic Wadge hierarchies. All results are based on a complete and
explicit description of the Wadge hierarchy on an arbitrary zero-dimensional
Polish space, depending on its topological properties.
