The present paper reviews author's studies on the dynamic vibration absorber in light of distinctive design criteria. Firstly, the exact solution to the min-max optimization of resonance amplitudes is described. The fixed points theory is a well-know classic approximate method for this problem. A numerical method is given for the damped primary system. This is followed by the definitions of the stability maximization and the variance minimization. The exact solutions are given for these criteria, and their characteristics are analyzed. The passive gyroscopic damper is referred to in connection with the similarity as the vibration absorbers and with the complementarity on the design optimizations.