  A combinatorial approach to free probability theory has been developped by
Roland Speicher, based on the notion of noncrossing cumulants, a free analogue
of the classical theory of cumulants in probability theory. We review this
theory, and explain the connections between free probability theory and random
matrices. We relate noncrossing cumulants to classical cumulants and also to
characters of large symmetric groups. Finally we give applications to the
asymptotics of representations of symmetric groups, specifically to the
Littlewood-Richardson rule.
