Littlewood-Offord theory is the study of random signed sums of n integers (or more generally, vectors), being particularly concerned with the probability that such a sum equals a fixed value (such as zero) or lies in a fixed set (such as the unit ball). Inverse Littlewood-Offord theory starts with some information about such probabilities (e.g. that a signed sum equals 0 with high probability) and deduces structural information about the original spacings (typically, that they are largely contained within a progression). We give examples of such theorems and describe some of the applications to random matrix theory. Our focus will be on the simplest applications, rather than the most recent ones.