A review is given on the progress in the study of general solutions of elasticity and their applications since 1972. Apart from summarizing and remarking the development of the general solution method in literature, this review aims to present the readers with a systematic and constructive scheme to develop general solutions from given governing differential equations and then to prove their completeness and investigate their nonuniqueness features. The effectiveness of the constructive scheme manifests itself in the fact that almost all the classic solutions, including not just classic displacement potentials but also classic stress functions, can be rederived by using this scheme. Furthermore, thanks to the systematic features of the scheme, it produces a constructive approach to study the completeness and nonuniqueness of general solutions and possesses more flexibility, which facilitates the extension of elastic general solution methods to more general systems governed by elliptic differential equations. Under the framework of this scheme, a comprehensive review is presented on wide application of general solutions in a variety of research areas, ranging from problems with different materials, isotropic or anisotropic, to various coupling problems, such as thermoelasticity, magnetoelasticity, piezoelectric elasticity, porous elasticity, and quasicrystal elasticity, and to problems of different engineering structures, for instance, the refined theories for beams and plates. There are 213 references cited in this review article.