Linear-time algorithms have long been considered the gold standard of computational efficiency. Indeed, it is hard to imagine doing better than that, since for a nontrivial problem, any algorithm must consider all of the input in order to make a decision. However, as extremely large data sets are pervasive, it is natural to wonder what one can do in {em sub-linear} time. Over the past two decades, several surprising advances have been made on designing such algorithms. We will give a non-exhaustive survey of this emerging area, highlighting recent progress and directions for further research.