Wadge hierarchy on general topological spaces

The Wadge preorder is a tool to compare the complexity of subsets of topological spaces: if $A,B$ are subsets of the topological spaces $X,Y$, respectively, $A$ \emph{Wadge reduces} to $B$ if there exists a continuous function $f:A\to B$ such that $A=f^{-1}(B)$. While most of the earlier work on the Wadge preorder concerned zero-dimensional Polish spaces, recent investigations have involved more general kinds of spaces. This talk surveys some of the results and presents a few open problems and perspectives in the field.