From the longest increasing subsequence in a random permutation to the shortest distance in a randomly weighted two dimensional Euclidean lattice, a large class of planar random growth models are believed to exhibit shared large scale features of the so-called Kardar-Parisi-Zhang (KPZ) universality class. Over the last 25 years, intense mathematical activity has led to a lot of progress in the understanding of these models, and connections to several other topics such as algebraic combinatorics, random matrices and partial differential equations have been unearthed. I shall try to give an elementary introduction to this area, describe some of what is known as well as many questions that remain open.