Integrable fluctuations in the KPZ universality class

The KPZ fixed point is a scaling invariant Markov process which arises as the universal scaling limit of a broad class of models of random interface growth in one dimension, the one-dimensional KPZ universality class. In this survey we review the construction of the KPZ fixed point and some of the history that led to it, in particular through the exact solution of the totally asymmetric simple exclusion process, a special solvable model in the class. We also explain how the construction reveals the KPZ fixed point as a stochastic integrable system, and how from this it follows that its finite dimensional distributions satisfy a classical integrable dispersive PDE, the Kadomtsev-Petviashvili (KP) equation.