This paper proposes a predictive and nonlinear robust control strategy to solve the path following (PF) problem for under-actuated quadrotors. However, most state-of-the-art methods fail to consider additional requirements, such as speed regulation on the path and internal stability such that quadrotors cannot further stabilize a desired point along the path. To address this problem, we firstly utilize transverse feedback linearization (TFL) to transform the original PF description in the output space into a path manifold description in the state space. Then, we design a model predictive path following control (MPFC) scheme for the outer-loop controller design, which generates the reference roll and pitch angles. The inner-loop controller is designed to track these reference angles by using a sliding mode control (SMC) in the presence of model uncertainties and disturbance. Finally, comparison studies are given to illustrate the effectiveness and improvement of proposed control strategy.