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Techniques are developed to calculate the energy production in quantum fields which obtain a mass through the spontaneous symmetry breaking of a second field which is undergoing a phase transition. All fields are assumed to be out of thermal equilibrium and weakly coupled. The energy produced in a field, which is initially in its ground state, is computed for two generic types of timedependent masses: a roughly monotonic turn on of the mass and an oscillatory mass. The formalism is applied to the questions of particle production and reheating in inflationary universe models. Requirements are found which the couplings in new-inflation-type models must satisfy for efficient reheating to occur.