Kardar, Parisi and Zhang introduced a universality class (the so-called KPZ universality class) in 1986 which is believed to explain the universal behaviour in a large class of two dimensional random growth models including first and last passage percolation. A number of breakthroughs has led to an explosion of mathematically rigorous results in this field in recent years. However, these have mostly been restricted to the class of "exactly solvable models", where exact formulae are available using powerful tools of random matrices, algebraic combinatorics and representation theory; beyond this class the understanding remains rather limited. I shall talk about a geometric approach to these problems based on studying the geometry of geodesics (optimal paths), and describe some recent progress along these lines including the resolution of Lebowitz's longstanding slow bond problem.
