An analysis is performed to investigate free and forced oscillation of a gas bubble and cavitation in viscoelastic liquids of a three‐parameter Oldroyd model. The first‐order perturbation method for small amplitudes has been employed to obtain periodic solutions to the bubble dynamics equation. Consideration is given to the influence of surface tension, vapor pressure, and thermodynamic behavior of the gas inside the bubble. It is disclosed that transient behavior of the bubble in viscoelastic liquids is governed by five dimensionless parameters. Conditions for stable bubbles are obtained. Free oscillation consists of a decaying exponential component and a damped sinusoidal oscillation associated with the natural frequency of the bubble‐liquid system. The effects of the five governing parameters on the natural frequency are determined. The radial motion of the bubble in an oscillating pressure field includes a sinusoidal oscillation associated with the forcing frequency, a decaying exponential component, and a damped sinusoidal oscillation associated with the natural frequency. Criteria for the onset of incipient cavitation are determined when a bubble is suddenly released into the liquid or is situated in an oscillating pressure field.