We describe the evolution of double degenerate binary systems, consisting of components obeying the zero-temperature mass-radius relationship for white dwarf stars, from the onset of mass transfer to one of several possible outcomes, including merger, tidal disruption of the donor, or survival as a semidetached AM CVn system. We use a combination of analytic solutions and numerical integrations of the standard orbit-averaged first-order evolution equations, including direct-impact accretion and the evolution of the components due to mass exchange. We include also the effects of mass loss during supercritical (super-Eddington) mass transfer and the tidal and advective exchanges of angular momentum between the binary components. With the caveat that our formalism does not include an explicit treatment of common-envelope phases, our results suggest that a larger fraction of detached double white dwarfs survive the onset of mass transfer than has been hitherto assumed, even if this mass transfer is initially unstable and rises to super-Eddington levels. In addition, as a consequence of the tidal coupling, systems that come into contact near the mass transfer instability boundary undergo a phase of oscillation cycles in their orbital period (and other system parameters). Unless the donor star has a finite entropy such that the effective mass-radius relationship deviates significantly from that of a zero-temperature white dwarf, we expect our results to be valid. Much of the formalism developed here would also apply to other mass-transferring binaries, and in particular to cataclysmic variables and Algol systems.