Conventional diagnostic ultrasound images of the anatomy (as opposed to blood flow) reveal differences in the acoustic properties of soft tissues (mainly echogenicity but also, to some extent, attenuation), whereas ultrasound-based elasticity images are able to reveal the differences in the elastic properties of soft tissues (e.g., elasticity and viscosity). The benefit of elasticity imaging lies in the fact that many soft tissues can share similar ultrasonic echogenicities but may have different mechanical properties that can be used to clearly visualize normal anatomy and delineate pathologic lesions. Typically, all elasticity measurement and imaging methods introduce a mechanical excitation and monitor the resulting tissue response. Some of the most widely available commercial elasticity imaging methods are 'quasi-static' and use external tissue compression to generate images of the resulting tissue strain (or deformation). In addition, many manufacturers now provide shear wave imaging and measurement methods, which deliver stiffness images based upon the shear wave propagation speed. The goal of this review is to describe the fundamental physics and the associated terminology underlying these technologies. We have included a questions and answers section, an extensive appendix, and a glossary of terms in this manuscript. We have also endeavored to ensure that the terminology and descriptions, although not identical, are broadly compatible across the WFUMB and EFSUMB sets of guidelines on elastography (Bamber et al. 2013; Cosgrove et al. 2013).