Results are presented of integrations of a representative sample of thousands of test bodies in the Kuiper belt for times up to the age of the solar system. Using this survey, the main regions of dynamical stability (and instability) are mapped out as a function of the particles' semi-major axes, eccentricities, and inclinations. Much of the dynamical structure is shown to be correlated with mean motion and secular resonances. Weak dynamical instabilities are confirmed to be capable of producing an influx of Neptune-approaching bodies even on Gyr time scales. The integrations are also compared with current observations of bodies in this region. Implications for the early evolution of the Kuiper belt, its current structure, and the origins of short-period comets are discussed.