Hyper-elasticmaterials are a kind of irreplaceable macromolecule materials withunique characteristics and wide applications. Researches on theunique instability problems of hyper-elastic materials havegreatly promoted the development of finite deformation andhyper-elastic theory in the field of continuum mechanics. Variousinstability problems for hyper-elastic materials are reviewed inthis paper, such as the Rivlin cube problem, inflation ofthin-walled cylindrical tubes and thin-walled spherical shells,torsional instability of cylindrical tubes, surface instability ofcompressed or bent blocks, the formation of void and the suddengrowth and collapse of void. The characteristics and solutionmethods are then analyzed for this nonlinear problem of materialinstability, and finally, main results and future developments aregiven and discussed.