We have numerically investigated the long-term dynamical behavior of known Centaurs. This class of objects is thought to constitute the transitional population between the Kuiper belt and the Jupiter-family comets (JFCs). In our study, we find that over their dynamical lifetimes these objects diffuse into the JFCs and other sinks, and they also make excursions into the scattered disk, but (not surprisingly) do not diffuse into the parameter space representing the main Kuiper belt. These Centaurs spend most of their dynamical lifetimes in orbits of eccentricity 0.2–0.6 and perihelion distance 12–30 AU. Their orbital evolution is characterized by frequent close encounters with the giant planets. Most of these Centaurs will escape from the solar system (or enter the Oort cloud), while a fraction will enter the JFC population and a few percent will impact a giant planet. Their median dynamical lifetime is 9 Myr, although there is a wide dispersion in lifetimes, ranging from less than 1 Myr to more than 100 Myr. We find the dynamical evolution of this sample of Centaurs to be less orderly than the planet-to-planet "handoff" described in previous investigations. We discuss the implications of our study for the spatial distribution of the Centaurs as a whole.