Abstract: One-dimensional interfaces growing in time (consider, for example, the top envelope of the configuration in the game of TETRIS) are ubiquitous in nature. I shall describe a class of stochastic models for interface growth that are believed to, asymptotically, share the same universal characteristics observed in many naturally occurring interfaces, and sketch, in parts, an ongoing story of the fascinating mathematics developed over the last twenty years with a view to understand such interfaces rigorously.