Consider a random sum η1v1+…+ηnvn, where η1,…,ηn are independently and identically distributed (i.i.d.) random signs and v1,…,vn are integers. The Littlewood-Offord problem asks to maximize concentration probabilities such as ℙ(η1v1+…+ηnvn=0) subject to various hypotheses on v1,…,vn. In this paper we develop an inverse Littlewood-Offord theory (somewhat in the spirit of Freiman’s inverse theory in additive combinatorics), which starts with the hypothesis that a concentration probability is large, and concludes that almost all of the v1,…,vn are efficiently contained in a generalized arithmetic progression. As an application we give a new bound on the magnitude of the least singular value of a random Bernoulli matrix, which in turn provides upper tail estimates on the condition number.