The periodic shedding of vortices that accompanies the cross flow past a bluff cylindrical body can excite the body into resonant transverse oscillations when the vortex shedding and body natural frequencies are sufficiently near to one another. A mathematical model that enables one to predict the vortex-excited resonant response of bluff cylinders is introduced here. A modified Van der Pol equation is employed as the governing equation for the fluctuating lift on the cylinder and is coupled to the equation for the oscillatory motion of the body. When appropriate choices are made for the empirical parameters in the model, the calculated responses of four spring mounted systems are in good quantitative agreement with the observed responses from wind tunnel experiments. A set of relations is postulated between the empirical parameters and the physical mass and damping parameters that govern the oscillatory response. These relations are then employed with the model to calculate the vortex-excited responses of several other systems. Good quantitative agreement with the measured data is again obtained.