Solar eruptions, which generate various types of wave phenomena in the solar atmosphere, are sources of space weather perturbations. These waves not only transport a large amount of energy, but also illuminate the properties of the magnetic field and plasma on the path of propagation. Therefore, it is of great interest to investigate wave phenomena in the solar atmosphere, among which Moreton waves in the solar chromosphere and extreme ultraviolet (EUV) waves in the corona have attracted much attention in the past few decades. Moreton waves are characterized by bright fronts in the Hα line center and blue wing (or dark fronts in the Hα red wing), which propagate at speeds ranging from ~500 km/s to more than 2000 km/s. They were also observed in He I 10830 Å. EUV waves are characterized by bright fronts in the EUV images, which propagate at speeds ranging from ~10 km/s to more than 2000 km/s. Whereas the understanding of Moreton waves is rather mature, the nature of EUV waves and their relationship with Moreton waves are controversial and have been debated for more than two decades. Initially, it was proposed that EUV waves are the coronal counterparts of chromospheric Moreton waves; that is, they are fast-mode MHD waves. However, many EUV waves have been found to have speeds that are less than the sound speed, which means that some EUV waves cannot be accounted for by the fast-mode MHD wave model. Therefore, several alternate models have been proposed, such as slow-mode soliton waves, echoes of fast-mode waves, successive magnetic reconnection models, as well as hybrid models, which predict the existence of two components of EUV waves—a fast-mode and a slower pseudo-wave. The pseudo-wave is explained by the magnetic field line stretching model, which predicts that the fast-mode wave should be ~3 times faster than the pseudo-wave. With later high-cadence observations, EUV waves with two components have been revealed in many events, confirming the validity of the hybrid model. However, recent observations have revealed many new features that deserve further elaboration of the existing models, such as secondary EUV waves, small-scale EUV waves, patchy EUV waves, quasi-periodic EUV waves, multiple EUV waves, homologous EUV waves, and stationary EUV waves. In particular, the very recent observations of stationary EUV waves may indicate that the fast component of the EUV wave might experience a mode conversion from fast to slow mode when the wave crosses a region of weak magnetic field. In this article, we review the progress made in the research of Moreton and EUV waves and discuss in detail the discovery of the Moreton wave, the early classical model of Moreton waves (as well as recent modifications), the discovery of EUV waves, various observational features, and various models of EUV waves. Lastly, we offer our perspectives on the current research in Moreton and EUV waves and highlight the importance of this research.