The skeleton of an isolated null point in three dimensions consists of a `spine curve' and a `fan surface'. Two isolated magnetic field lines approach (or recede from) the null point from both directions along the spine, and a continuum of field lines recedes from (or approaches) the null in the plane of the fan surface. Two bundles of field lines approach the null point around the spine (one from each direction) and spread out near the fan. The kinematics of steady reconnection at such a null point is considered, depending on the nature of the imposed boundary conditions on the surface that encloses the null, in particular on a cylindrical surface with its axis along the spine. Three kinds of reconnection are discovered. In `spine reconnection' continuous footpoint motions are imposed on the curved cylindrical surface, crossing the fan and driving singular jetting flow along the spine. In `fan reconnection' continuous footpoint motions are prescribed on the ends of the cylinder, crossing the spine and driving a singular swirling motion at the fan. An antireconnection theorem is proved, which states that steady MHD reconnection in three dimensions with plasma flow across the spine or fan is impossible in an inviscid plasma with a highly subAlfvenic flow and uniform magnetic diffusivity. One implication of this is that reconnection tends to be an inherently nonlinear phenomenon. A linear theory for slow steady reconnection is developed which demonstrates explicitly the nature of the spine singularity in spine reconnection. Finally, the properties of `separator reconnection' in complex configurations containing two null points are discussed by means of analytical examples.