We present PANOC, a new algorithm for solving optimal control problems arising in nonlinear model predictive control (NMPC). A usual approach to this type of problems is sequential quadratic programming (SQP), which requires the solution of a quadratic program at every iteration and, consequently, inner iterative procedures. As a result, when the problem is ill-conditioned or the prediction horizon is large, each outer iteration becomes computationally very expensive. We propose a line-search algorithm that combines forward-backward iterations (FB) and Newton-type steps over the recently introduced forward-backward envelope (FBE), a continuous, real-valued, exact merit function for the original problem. The curvature information of Newton-type methods enables asymptotic superlinear rates under mild assumptions at the limit point, and the proposed algorithm is based on very simple operations: access to first-order information of the cost and dynamics and low-cost direct linear algebra. No inner iterative procedure nor Hessian evaluation is required, making our approach computationally simpler than SQP methods. The low-memory requirements and simple implementation make our method particularly suited for embedded NMPC applications.