We argue that light sails with nanometer-scale thicknesses that are rapidly accelerated to relativistic velocities by lasers must be significantly curved in order to reduce their intrafilm mechanical stresses and avoid tears. Using an integrated opto-thermo-mechanical model, we show that the diameter and radius of curvature of a circular light sail should be comparable in magnitude, both on the order of a few meters, in optimal designs for gram-scale payloads. Moreover, we demonstrate that, when sufficient laser power is available, a sail’s acceleration length decreases as its curvature increases. We will also discuss initial efforts to experimentally realize some of the sail structures with candidate materials that simultaneously fit the mass budget and reflectivity constraints. Our findings provide critical guidance for emerging light sail design programs, which herald a new era of interstellar space exploration to destinations such as the Oort cloud, the Alpha Centauri system, and beyond.