The astounding classical theorem of Haar/Weil/Cartan states that every locally compact topological group admits a Haar measure, i.e. a translation-invariant regular Borel measure. Thus for locally compact groups, we have a canonical measure-theoretic notion of "small" and "large" sets, which serve as an analogue for the purely topological concept of Baire category. But what about for non-locally compact groups, which generally do not support regular Borel measures in a natural way? In this talk I will describe the family of sets which are sometimes called "Haar null sets," sometimes "shy sets," which are defined measure-theoretically in any complete metric group $G$. I will demonstrate that the family of Haar null sets is a $\sigma$-ideal if $G$ is separable, and that it coincides with the family of Haar measure zero sets if $G$ is locally compact. I will give a detailed history of the subject and present some new results and open problems.