This paper presents a real-time capable solution method for nonconvex trajectory optimization with continuous-time constraint satisfaction and guaranteed convergence. The method, called successive convexification, combines several key techniques: exterior penalty-based reformulation of path constraints, generalized time-dilation, multiple-shooting discretization, $\ell_1$ exact penalization of nonconvex constraints, and the prox-linear method for convex-composite minimization. The reformulation of path constraints enables continuous-time constraint satisfaction even on sparse discretization grids, eliminating the need for mesh refinement heuristics. The prox-linear method guarantees convergence to stationary points of the penalized problem and ensures that converged solutions are Karush-Kuhn-Tucker (KKT) points. The method is demonstrated on numerical examples involving dynamic obstacle avoidance and rocket landing. The framework is shown to be effective and real-time capable, with the ability to handle both convex and nonconvex optimal control problems. The method is particularly useful for applications where continuous-time feasibility is essential, such as autonomous systems in aerospace and robotics.This paper presents a real-time capable solution method for nonconvex trajectory optimization with continuous-time constraint satisfaction and guaranteed convergence. The method, called successive convexification, combines several key techniques: exterior penalty-based reformulation of path constraints, generalized time-dilation, multiple-shooting discretization, $\ell_1$ exact penalization of nonconvex constraints, and the prox-linear method for convex-composite minimization. The reformulation of path constraints enables continuous-time constraint satisfaction even on sparse discretization grids, eliminating the need for mesh refinement heuristics. The prox-linear method guarantees convergence to stationary points of the penalized problem and ensures that converged solutions are Karush-Kuhn-Tucker (KKT) points. The method is demonstrated on numerical examples involving dynamic obstacle avoidance and rocket landing. The framework is shown to be effective and real-time capable, with the ability to handle both convex and nonconvex optimal control problems. The method is particularly useful for applications where continuous-time feasibility is essential, such as autonomous systems in aerospace and robotics.