We propose a UV mapping algorithm that jointly optimizes for cuts and distortion, sidestepping heuristics for placing the cuts. The energy we minimize is a state-of-the-art geometric distortion measure, generalized to take seams into account. Our algorithm is designed to support an interactive workflow: it optimizes UV maps on the fly, while the user can interactively move vertices, cut mesh parts, join seams, separate overlapping regions, and control the placement of the parameterization patches in the UV space. Our UV maps are of high quality in terms of both geometric distortion and cut placement, and compare favorably to those designed with traditional modeling tools. The UV maps can be created in a fraction of the time as existing methods, since our algorithm drastically alleviates the trial-and-error, iterative procedures that plague traditional UV mapping approaches.