In this paper, a parameterized NMPC scheme is presented for path following and stabilization of a quadrotor helicopter. The control strategy deals structurally with the constraints on control inputs and ensures nice performance for attitude and position control. Besides, the resulting parametrization decreases substantially the number of decision variables related to the nonlinear optimization problem which hugely facilitates the attainment of the optimal control sequence. Moreover, the NMPC framework enables the use of different nonlinear formulations of the quadrotor whatever the complexity of the model being used. Simulation results are shown in order to highlight the advantages of the proposed control scheme.