As part of a program on noncommutative Laurent phenomenon, we introduce and study noncommutative Catalan "numbers" (as Laurent polynomials in infinitely many free variables) and related theory of noncommutative binomial coefficients. We also study their specializations, both commutative and noncommutative; relate them with Garsia-Haiman (q,t)-versions; and establish total positivity of the associated Hankel matrices. This is joint work with Arkady Berenstein (University of Oregon). Speaker(s): Vladimir Retakh (Rutgers U.)