Kardar-Parisi-Zhang universality class: exactly solvable models and universal laws

In 1986, Cardar, Parisi, and Zhang tried to find a universal description of the growth of surfaces under the action of random forces and proposed an eponymous equation. It turned out that the behavior of solutions to this equation is inherent to many phenomena of different nature, which are now combined into the Kardar-Parisi-Zhang (KPZ) class of universality. The beginning of the twenty-first century was marked by a breakthrough in the understanding of physics and mathematics of these phenomena in one dimension. I will tell about this progress and review some problems, the solution of which helps one to understand the nature of the KPZ universality class.