Abstract: Since Love (1911) studied the elastic deformation of an elastic self-gravitation sphere, many scientists have carried out in-depth research on seismic deformation problems in different viscoelastic Earth models, and mainly developed related theories based on semi-infinite space and spherical Earth model. The problem of seismic deformation is usually treated by integral transformation, basis function expansion, and other techniques, and is simplified to solve ordinary differential equations satisfying the boundary conditions at the source and on the Earth's surface. In view of this unique boundary value problem in mathematical physics, this paper summarizes various theoretical methods developed in recent decades in the form of full-analytical, semi-analytical and numerical integral solutions, and discusses the characteristics of various methods. In addition, the progress and problems of the 3-D Earth models are briefly reviewed. In a word, this paper summarizes the development history, research status, and latest progress of the deformation theory in viscoelastic Earth, in the past half-century, and discusses the existing problems and future development trend.