In the last third of the 20th century probability theory branched out from the study of stochastic processes indexed by one-dimensional time to the study of stochastic models with new types of complex spatial dependencies.   Examples of this include random growth models, random trees and graphs, percolation, random paths that interact with their environment, and interacting particle systems.  This talk takes a look at random growth models. We describe some of the successes that have been won over the last 50 years, and some of the myriad challenges that remain. These themes are discussed through a central  example known as the corner growth model. Classical random walk leads us to the basic questions studied in probability theory and serves as a useful counterpoint to the features of random growth.