An advanced ascent guidance algorithm for rocket-powered launch vehicles is developed. The ascent guidance function is responsible for commanding attitude, throttle and setting during the powered ascent phase of flight so that the vehicle attains target cutoff conditions in a near-optimal manner while satisfying path constraints such as maximum allowed bending moment and maximum allowed axial acceleration. This algorithm cyclically solves the calculus-of-variations two-point boundary-value problem starting at vertical rise completion through orbit insertion. This is different from traditional ascent guidance algorithms which operate in an open-loop mode until the high dynamic pressure portion of the trajectory is over, at which time there is a switch to a closed loop guidance mode that operates under the assumption of negligible aerodynamic forces. The main contribution of this research is an algorithm of the predictor-corrector type wherein the state/costate system is propagated with known (navigated) initial state and guessed initial costate to predict the state/costate at engine cutoff. The initial costate guess is corrected, using a multi-dimensional Newton?s method, based on errors in the terminal state constraints and the transversality conditions. Path constraints are enforced within the propagation process. A modified multiple shooting method is shown to be a very effective numerical technique for this application. Results for a single stage to orbit launch vehicle are given. In addition, the formulation for the free final time multi-arc trajectory optimization problem is given. Results for a two-stage launch vehicle burn-coast-burn ascent to orbit in a closed-loop guidance mode are shown. An abort to landing site formulation of the algorithm and numerical results are presented. A technique for numerically treating the transversality conditions is discussed that eliminates part of the analytical and coding burden associated with optimal control theory.