We consider situations where a nonlinear dynamical system possesses a smooth invariant manifold. For parameter values p less than a critical value p_c, the invariant manifold has within it a chaotic attractor of the system. As p increases through p_c a blowout bifurcation takes place, in which the former attraction to the manifold changes to repulsion, and the chaotic set in the manifold ceases to be an attractor of the system. Depending on the dynamics away from the manifold, blowout bifurcations can be either hysteretic or nonhysteretic, and they are correspondingly accompanied either by riddled basins (in which the basin is a “fat fractal”) or by an extreme form of temporally intermittent bursting recently called on-off intermittency. The role of the dynamics away from the manifold in determining the hysteretic or supercritical nature of the bifurcation is explicitly illustrated with a numerical example.