The KPZ equation, a model for the random growth of rough interfaces, has been the subject of great physical and mathematical interest since its introduction in 1986. By simple changes of variables, it is closely related to the stochastic heat equation, which models the partition function of a random walk in a random environment, and the stochastic Burgers equation, a simple model for turbulence. I will explain several recent results about the existence, classification, and properties of spacetime-stationary solutions to these equations on ${\bf R}^d$ for various values of $d$. These solutions thus represent the behavior of the models in large domains on long time scales. Most of the results are joint work with various combinations of C. Graham, Y. Gu, L. Ryzhik, and O. Zeitouni.