Indentation, owing to its ability to induce permanent deformation in a controlled manner even in highly brittle materials that are otherwise not amenable to critical mechanical testing experiments, is widely used for studying the mechanical behavior of amorphous solids. In this paper, an overview of the current understanding on the indentation response of inorganic glasses - both metallic and non-metallic - is presented. The stresses underneath a sharp indenter, such as the Vickers indenter that is most commonly used to evaluate hardness of glasses, can be substantial, which, in turn, can induce densification, plastic deformation, viscous flow, and cracking, depending on a number of material and experimental variables. In this paper, the physics and mechanics of these processes, the interplay between them, and their effect on hardness of glasses are reviewed. The effect of composition in various glass systems such as chalcogenides, phosphates, silicates, and silicon oxynitrides, on hardness are discussed. Different microcrack patterns that occur during indentation and fracture resistance of glasses are summarized. The effects of extrinsic parameters such as time, temperature, and pressure on the deformation behavior of glasses are highlighted. The use of the bonded-interface and nanoindentation techniques for studying incipient as well as the localized nature of plastic flow and its pressure sensitivity, and hardness in metallic glasses is summarized. A review of the nanoindentation's contribution for investigating the strain rate, ion irradiation, and environmental effects is provided. Finally, possible future avenues for further exploration of indentation of glasses are discussed.