Robotic simulation, planning, estimation, and control, have all been built on top of numerical optimization. In this same time, modern convex optimization has matured into a robust technology that delivers globally optimal solutions in polynomial time. With advances in differentiable optimization and custom solvers capable of producing smooth derivatives, convex modeling has become fast, reliable, and fully differentiable. This thesis demonstrates the effectiveness of convex modeling in areas such as Martian atmospheric entry guidance, nanosatellite space telescope pointing, collision detection, contact dynamics of point clouds, online model learning, and finally, a derivative-free method for trajectory optimization that leverages parallelized simulation. In all of these domains, the reliability and speed of differentiable convex optimization enables real-time algorithms that are rigorous, performant, and easy to understand and modify.