High-cadence space-based observations, available for over a decade now, have revealed globally propagating wave-like disturbances in the solar corona. These coronal waves have now been imaged in a wide range of spectral channels, yielding a wealth of information. Still, no consensus on their physical nature has been reached yet. While many findings are consistent with fast-mode MHD waves and/or shocks, other characteristics have given rise to alternative models which involve magnetic reconfiguration in the framework of an erupting coronal mass ejection. In this paper, the observational signatures of coronal waves will be reviewed, and the different physical interpretations of coronal waves and how they are motivated by observations will be discussed. Finally, the potential of using coronal waves as a diagnostic tool for the corona will be shown.