This paper considers the problem of designing predictive control laws for nonlinear autoregressive exogenous(NARX) systems based on measured input/output data without explicitly identifying the parameters of the system.First, we consider the nominal (noise-free) case, where we prove the recursive feasibility and stability of the closed-loop system. Then, we explore the case when outputs are corrupted by additive measurement noise. Finally, we discuss the case of capturing the real-time deviations in the system by adapting the data Hankel matrices. In this respect, to guarantee the persistency of excitation condition, we consider three triggering mechanisms, namely, absolute, relative, and mixed. We have illustrated all the discussed cases with examples.