Global extreme ultraviolet (EUV) waves are spectacular traveling disturbances in the solar corona associated with energetic eruptions such as coronal mass ejections (CMEs) and flares. Over the past 15 years, observations from three generations of space-borne EUV telescopes have shaped our understanding of this phenomenon and at the same time led to controversy about its physical nature. Since its launch in 2010, the Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory (SDO) has observed more than 210 global EUV waves in exquisite detail, thanks to its high spatio-temporal resolution and full-disk, wide-temperature coverage. A combination of statistical analysis of this large sample, more than 30 detailed case studies, and data-driven MHD modeling, has been leading their physical interpretations to a convergence, favoring a bimodal composition of an outer, fast-mode magnetosonic wave component and an inner, non-wave CME component. Adding to this multifaceted picture, AIA has also discovered new EUV wave and wave-like phenomena associated with various eruptions, including quasi-periodic fast propagating (QFP) wave trains, magnetic Kelvin-Helmholtz instabilities (KHI) in the corona and associated nonlinear waves, and a variety of mini-EUV waves. Seismological applications using such waves are now being actively pursued, especially for the global corona. We review such advances in EUV wave research focusing on recent SDO/AIA observations, their seismological applications, related data-analysis techniques, and numerical and analytical models.