Techniques to measure and manipulate the carrier-envelope phase within femtosecond optical parametric oscillators (OPOs) allow their outputs to be stabilized in a way that produces a frequency comb structure, potentially tunable throughout the transparency band of the gain material. In this review we describe the fundamental principles of phase control, on which the development of singly- and doubly-resonant OPO frequency combs is based. We give examples of practical embodiments of such combs, and discuss in detail several applications, including spectroscopy, metrology, quantum computation and astrophotonics.