We present an algorithmic framework for solving parametric nonlinear optimization problems. It builds on the successive solution of convex optimization problems that are obtained by linearizing nonlinear equality constraints and keeping all convex structures of the problem intact. A special case of the scheme is the constrained Gauss-Newton method. The complete scheme might be might be characterized as adjoint-based predictor-corrector sequential convex programming. After presenting the algorithm, we prove a contraction estimate that guarantees the tracking performance of the algorithm with respect to the exact solutions of a family of parameter varying problems. The scheme can used to treat online parametric nonlinear programming problems as they arise in nonlinear model predictive control (NMPC). We report on recent algorithmic advances to speed up the NMPC computations by automatic code generations and present simulated applications and report on experiments that have been obtained at with control of a small tractor, an overhead crane, and fast flying tethered airfoils, all with computation times of 1-5 milliseconds per optimal control problem. The talk presents joint work with Quoc Tran Dinh and Carlo Savorgnan.