The Catalan numbers and their generalizations and refinements (Fuss numbers, Cayley-Kirkman numbers, Narayana numbers, etc.) can be viewed as "type A" versions of more general numbers defined for an arbitrary finite Coxeter group. These numbers come up in a variety of combinatorial, algebraic, and geometric contexts to be surveyed in the talk (hyperplane arrangements, noncrossing partitions, generalized associahedra, and so on), suggesting connections that transcend mere numerology. Combinatorics of generalized Catalan numbers can be applied to the following problem. The Coxeter-Dynkin diagram of a finite root system determines its classical invariants such as the Coxeter number and the exponents. Can one recover these invariants directly from the diagram, without the intermediate step of constructing a root system or Coxeter group? (This is joint work with N.Reading.)