Message passing is a fundamental technique for performing calculations on networks and graphs with applications in physics, computer science, statistics, and machine learning, including Bayesian inference, spin models, satisfiability, graph partitioning, network epidemiology, and the calculation of matrix eigenvalues. Despite its wide use, however, it has long been recognized that the method has a fundamental flaw: It works poorly on networks that contain short loops. Loops introduce correlations that can cause the method to give inaccurate answers or to fail completely in the worst cases. Unfortunately, most real-world networks contain many short loops, which limits the usefulness of the message-passing approach. In this paper we demonstrate how to rectify this shortcoming and create message-passing methods that work on any network. We give 2 example applications, one to the percolation properties of networks and the other to the calculation of the spectra of sparse matrices.