Structural vibrations are often dominated by resonant response, and increased efficiency in the damping of these vibrations can often be attained by using the resonant properties of these modes. A typical example is the ‘tuned mass absorber’. While the original design procedure was based on properties of the frequency response graph, it has recently been demonstrated that the problem can be generalized and solved by use of the complex root-locus properties. The basic principle in this formulation is the introduction of a resonant force with frequency tuning that results in splitting the original resonant mode into two modes with equal damping ratio. Here the basic principle of resonant absorbers is presented in concise form and generalized in two ways. First the principle of resonant absorbers is presented in a general form in terms of sensors and actuators, recording the motion and imposing appropriate forces, respectively. A general design procedure is developed for resonant displacement and acceleration feedback, respectively, based on a combination of ‘equal modal damping’ and approximately equal response amplitudes of the two modes. This leads to explicit design formulae for the parameters of the control system. The optimal calibration leads to a plateau of near-equal amplification in a frequency interval around the original natural frequency. In multi-degree-of-freedom structures the sensor picks up the total deformation from all the modes that are active at the location of the sensor, and the actuator force acts on all these modes. A quasi-static correction is developed that identifies the part of the sensor signal associated with the mode to be damped and the reduced effect of the actuator on this mode. This correction takes the form of explicit modifications of the formulae for the optimal control parameters, while retaining the original format. The efficiency of resonant damping is illustrated by application to a benchmark example for stochastic wind load on a high-rise building. It is demonstrated that the present resonant damping technique based on a single collocated sensor is competitive with more heavily instrumented configurations using the classic LQG technique.