The sequence 1, 1, 2, 5, 14, 42, ... of Catalan numbers is probably the most ubiquitous and most interesting sequence in combinatorics. For instance, the Online Encyclopedia of Integer Sequences says that the Catlan number entry A000108 "is probably the longest entry in the OEIS, and rightly so." I have recently completed a monograph on Catalan numbers, to be published by Cambridge University Press. I will give a sample of results from this monograph, including selections from the history of Catalan numbers, their "transparent" combinatorial interpretations, more subtle combinatorial interpretations, algebraic interpretations, and analytic and number-theoretic properties.