Risk scores are simple classification models that let users quickly assess risk by adding, subtracting, and multiplying a few small numbers. Such models are widely used in healthcare and criminal justice, but are often built ad hoc. In this paper, we present a principled approach to learn risk scores that are fully optimized for feature selection, integer coefficients, and operational constraints. We formulate the risk score problem as a mixed integer nonlinear program, and present a new cutting plane algorithm to efficiently recover its optimal solution. Our approach can fit optimized risk scores in a way that scales linearly with the sample size of a dataset, provides a proof of optimality, and obeys complex constraints without parameter tuning. We illustrate these benefits through an extensive set of numerical experiments, and an application where we build a customized risk score for ICU seizure prediction.