The focus of this review is on the basic idea and useful interpretations of reduction methods; multiple-parameter reduction methods; treatment of nonconservative loadings (unsymmetric systems); and application of reduction methods in conjunction with operator splitting. The literature reviewed is devoted to the mathematical aspects of reduction methods, as well as to the following seven application areas: eigenvalue problems; nonlinear vibrations; linear and nonlinear dynamic analysis (initial/boundary value problems); linear systems analyzed by using semi-analytic numerical discretization procedures; reanalysis techniques; sensitivity analysis; and optimum design. Sample numerical results are presented showing the effectiveness of reduction methods in a recent application. Hybrid analytical techniques which share some of the key elements with reduction techniques are highlighted. Some of the future directions for research on reduction methods are outlined.