Gusts of moderate and large magnitude induce flow separation and other complexities when they interact with the lifting surfaces of air vehicles. The presence of these nonlinear gusts are becoming ubiquitous in twenty-first-century air vehicles, where the classic potential flow-based methodologies applied in the past may no longer be valid. In this review, we define the parameter space for the presence of large-amplitude gusts and describe where and when these gusts may primarily be found. Recent research using modern experimental and computational techniques to define the limits of classical unsteady and indicial aerodynamic theories is summarized, with a focus on discrete transverse, streamwise (longitudinal), and vortex gust encounters. We propose areas where future research is needed to transition these studies of large-amplitude gust physics to real-time prediction and mitigation during flight.