The paper introduces a novel method called *successive convexification* for solving nonconvex trajectory optimization problems with continuous-time constraint satisfaction and guaranteed convergence. The method combines several key techniques: exterior penalty reformulation of path constraints, generalized time-dilation, multiple-shooting discretization, $\ell_1$ exact penalization of nonconvex constraints, and the prox-linear method, an SCP algorithm for convex-composite minimization. This approach ensures continuous-time feasibility even on sparse discretization grids and guarantees convergence to stationary points of the penalized problem. The paper also highlights the specialization of this property to global minimizers of convex optimal control problems, where the reformulated path constraints cannot be represented by canonical cones. The effectiveness and real-time capability of the proposed framework are demonstrated through numerical examples in dynamic obstacle avoidance and rocket landing applications.The paper introduces a novel method called *successive convexification* for solving nonconvex trajectory optimization problems with continuous-time constraint satisfaction and guaranteed convergence. The method combines several key techniques: exterior penalty reformulation of path constraints, generalized time-dilation, multiple-shooting discretization, $\ell_1$ exact penalization of nonconvex constraints, and the prox-linear method, an SCP algorithm for convex-composite minimization. This approach ensures continuous-time feasibility even on sparse discretization grids and guarantees convergence to stationary points of the penalized problem. The paper also highlights the specialization of this property to global minimizers of convex optimal control problems, where the reformulated path constraints cannot be represented by canonical cones. The effectiveness and real-time capability of the proposed framework are demonstrated through numerical examples in dynamic obstacle avoidance and rocket landing applications.