An optimization-based path-following control scheme for the lateral control of a nonholonomous vehicle is proposed. It is formulated as a nonlinear model predictive controller. The penalty function is chosen in a way that the result of the optimization is conceptually similar to the output of a traditional "Follow-the-Carrot" or "Pure Pursuit" approach as long as no constraints are active. This allows for intuitive tuning of the controller. However, as the controller is based on a nonlinear path prediction over several actuation steps, it can also anticipate e.g. sharp turns in the upcoming path that will cause the actuation variable (the steering angle) to saturate. Therefore, the controller can react earlier in case such a constraint is predicted to be run into. This results in very robust behavior with respect to the smoothness (drivability) of the reference path. The reference path can be given as a simple list of 2D coordinates, no explicit path planning is necessary. Advantages are seen especially when actuation limits (admissible lateral acceleration, maximum steering angle) are reached. The proposed controller is compared to a "Pure Pursuit" controller in simulation. Then, it is implemented on a model vehicle and experimental results are presented.