In the past two decades there has been considerable progress on understanding the asymptotic fluctuations of random growing interfaces.  These are often described by the Kardar-Parisi-Zhang equation, but this is a just one member of a huge universality class, and we have learned that special discretizations can have a high degree of solvability.  In the long-time, large-space limit, there is a new universal fixed point with unexpected connections to classical integrable systems.  The talk will be a gentle introduction to these developments.