Integrable probability has emerged as an active area of research at the
interface of probability/mathematical physics/statistical mechanics on the one
hand, and representation theory/integrable systems on the other. Informally,
integrable probabilistic systems have two properties: 1) It is possible to
write down concise and exact formulas for expectations of a variety of
interesting observables (or functions) of the system. 2) Asymptotics of the
system and associated exact formulas provide access to exact descriptions of
the properties and statistics of large universality classes and universal
scaling limits for disordered systems. We focus here on examples of integrable
probabilistic systems related to the Kardar-Parisi-Zhang (KPZ) universality
class and explain how their integrability stems from connections with symmetric
function theory and quantum integrable systems.Comment: Proceedings of the ICM, 31 pages, 10 figure