Human tissues must be elastic, much like other materials that work under continuous loads without losing functionality. The elasticity of tissues is provided by elastin, a unique protein of the extracellular matrix (ECM) of mammals. Its function is to endow soft tissues with low stiffness, high and fully reversible extensibility, and efficient elastic-energy storage. Depending on the mechanical functions, the amount and distribution of elastin-rich elastic fibers vary between and within tissues and organs. The article presents a concise overview of the mechanical properties of elastin and its role in the elasticity of soft tissues. Both the occurrence of elastin and the relationship between its spatial arrangement and mechanical functions in a given tissue or organ are overviewed. As elastin in tissues occurs only in the form of elastic fibers, the current state of knowledge about their mechanical characteristics, as well as certain aspects of degradation of these fibers and their mechanical performance, is presented. The overview also outlines the latest understanding of the molecular basis of unique physical characteristics of elastin and, in particular, the origin of the driving force of elastic recoil after stretching.