The performance characteristics of a continuously pumped doubly resonant optical parametric oscillator have been studied in detail. Using barium sodium niobate as the nonlinear material threshold for oscillation, as low as 3 mW has been observed. At higher pump power levels an efficiency exceeding 50 percent and a tunable output power of nearly 80 mW (signal plus idler), have been achieved. The effect of frequency stability of the pump on oscillator performance has been studied with single-mode operation and ±20-percent amplitude stability attained. The effects of cavity-length perturbations on oscillator performance have also been studied. It is found that cavity-length changes of wavelength (250 nm in this case) can produce tunability of the output frequency by as much as 100 cm-1. These results as well as their interpretation and implications on the design of doubly resonant optical parametric oscillators are discussed.