This paper presents a review of model predictive control and its applications in aircraft systems. Starting from representative mission scenarios and key technical challenges, it clarifies the design requirements for aircraft control systems. In response to the design needs of different classes of vehicles, the review surveys and synthesizes a coherent framework for model predictive control. It traces the origins and development of model predictive control and summarizes its theoretical foundations, with particular attention to robust model predictive control, Lyapunov-based model predictive control, switched model predictive control, and explicit model predictive control, thereby delineating the principal advances reported in recent years. Building on this framework, the paper examines applications of model predictive control to quadrotors, helicopters, fixed-wing aircraft, and high-speed aircraft. Finally, it outlines future research directions for model predictive control in aerospace control and offers concluding remarks.