The problem of steady state, incompressible magnetic reconnection in three dimensions is addressed. It is shown that exact reconnection solutions can be constructed by superposing nonlinear disturbances onto three-dimensional magnetic X-points. There are two distinct families of reconnection solutions. These can be understood in terms of the eigenstructure of the null, that is, in terms of the " spine" curves and "fan" surfaces that define the separatrices of the field. One family of solutions is driven by dis turbances in the fan and involves quasi-cylindrical current structures aligned to the axis of the spine; the other is associated with advection across the spine and a global current sheet aligned to the fan. Although both spine and fan solutions reduce to the two-dimensional analytic, shear-flow solutions of Craig & Henton, the three-dimensional spine current formulation allows far richer reconnective current structures.