We work out the maximal arithmetic subgroups of every orthogonal group G over the rational numbers (indefinite with dimension at least 3). The final answer is quite clean, but has some interesting exceptional cases. The local analysis has two main ingredients: first, new building-like complexes on which p-adic orthogonal groups act; second, a very detailed analysis of how the stabilizer of each vertex permutes its neighbors. The strong approximation theorem isn't quite enough to pass to the global case: for some quadratic forms over Z we must also construct reflections in their orthogonal groups. Our paper, giving all details, is available at https://web.ma.utexas.edu/users/allcock/research/omax12.pdf