This chapter gives an overview of recent developments of convexification and real-time convex optimization based control methods, in the context of Model Predictive Control (MPC). Lossless Convexification is a technique that formulates a class of non-convex control constraints as equivalent convex ones, while Successive Convexification gives an algorithm that targets nonlinear dynamics and certain non-convex state constraints. A large class of real-world optimal control problems can be solved with either method or a combination of both. For some time-critical applications, such as autonomous vehicles, it is crucial to have real-time capabilities. The real-time solution to these problems requires highly efficient customized convex programming solvers, which is also discussed as a part of this chapter. The effectiveness of convexification methods and real-time computation is demonstrated by a planetary soft landing problem throughout the chapter.