Catalan combinatorics of arbitrary finite type

The Catalan numbers and their q-analogues are known to count lots of different combinatorial objects. It then comes at no surprise that natural analogues of (q-) Catalan numbers that can be defined for any finite root system are just as ubiquitous. In particular, these generalized Catalan numbers have interpretations in terms of

- the Weyl group,
- the Lie group,
- the Lie algebra,
- the root lattice,
- the Coxeter arrangement,
- the braid group,
- the cluster algebra, and
- the associahedron

of the corresponding type. Some of these interpretations will be surveyed in the talk, which will be directed at a general mathematical audience.