Coxeter-Catalan Combinatorics

I will review some of the Catalan combinatorics associated to finite reflection groups: nonnesting objects for affine Weyl groups, noncrossing objects for well-generated finite complex reflection groups, and the recent construction using braid varieties of rational noncrossing objects for finite Coxeter groups. I will conclude with the still-open problem of finding an explicit uniform bijection between noncrossing and nonnesting objects.