Quasi-periodic, fast-mode propagating (QFP) wave trains in the corona have been studied intensively over the last decade, thanks to the full-disk, high spatio-temporal resolution, and wide-temperature coverage observations taken by the Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory (SDO). In the AIA observations, the QFP wave trains are seen to consist of multiple coherent and concentric wavefronts emanating successively near the epicenter of the accompanying flares. They propagate outwardly either along or across coronal loops at fast-mode magnetosonic speeds from several hundred to more than 2000 km s(-1), and their periods are in the range of tens of seconds to several minutes. Based on the distinctly different properties of QFP wave trains, they might be divided into two distinct categories: narrow and broad ones. For most QFP wave trains, some of their periods are similar to those of the quasi-periodic pulsations (QPPs) in the accompanying flares, indicating that they are probably different manifestations of the same physical process. Currently, candidate generation mechanisms for QFP wave trains include two main categories: the pulsed energy excitation mechanism associated with magnetic reconnection and the dispersion-evolution mechanism related to the dispersive evolution of impulsively generated broadband perturbations. In addition, the generation of some QFP wave trains might be driven by the leakage of three- and five-minute oscillations from the lower atmosphere. As one of the discoveries of SDO, QFP wave trains provide a new tool for coronal seismology to probe the corona parameters, and they are also useful for diagnosing the generation of QPPs, flare processes including energy release, and particle acceleration. This review aims to summarize the main observational and theoretical results of spatially resolved QFP wave trains in extreme-ultraviolet observations and presents briefly a number of questions that deserve further investigation.