A non-linear analysis is presented for the radial oscillations of a spherical gas bubble immersed in an inviscid, slightly compressible fluid. The mass of the gas bubble and the surface tension are assumed to be negligible. The radial velocity of the bubble is assumed to be small compared with the speed of sound. The response of the bubble to a plane, pressure-wave train whose wavelength is large compared with the radius of the bubble is determined by using the method of multiple scales when the frequency of the pressure wave (excitation frequency) is near the linear natural frequency of the bubble. The results show that the frequency response curves are similar to those obtained for a mass connected to a non-linear, soft spring. The peak, steady-state, response amplitude is proportional to the amplitude of the pressure wave, but occurs at a frequency slightly smaller than the linear natural frequency. The steady-state amplitude exhibits a jump phenomenon as the excitation frequency is swept up and down near the linear natural frequency. The peak amplitude is attainable for very special initial conditions. When the excitation frequency is approximately twice the linear natural frequency, subharmonic resonances take place and the motion is unstable if the excitation amplitude exceeds a critical value. On the other hand, when the excitation frequency is approximately one-half the linear natural frequency, the steady-state response is periodic for all values of the excitation amplitude.