A well tool apparatus for damping torsional vibration of a drill string comprises stabilizing members projecting radially outwards from a housing that is, in operation, rotationally integrated in the drill string, to stabilize the drill string by engagement with a borehole wall. The stabilizing members are displaceably mounted on the housing to permit limited angular movement thereof relative to the housing about its rotational axis. The well tool apparatus includes a hydraulic damping mechanism to damp angular displacement of the stabilizing members relative to the housing, thereby damping torsional vibration of the housing and the connected drill string, in use.