This paper investigates three-dimensional reconnection of magnetic fields with nulls, where B = 0, and of fields with closed field lines. We review the geometry of magnetic fields with nulls, focusing on the important structures associated with them, i.e., the γ lines and Σ surfaces. The geometric structures of configurations with a pair of type A and B nulls permit reconnection across the null-null lines, which are the field lines that join the two nulls. We also review magnetostatic reconnection, in which the specified magnetic field is time independent and the electrostatic potential φ is constant along field lines because of ideal Ohm's law E + v x B = 0. In this case, the problem reduces to mapping the potential along the field lines. We perform potential mapping on configurations with one or two nulls, identifying the essential singularities and step function discontinuities in the potential φ, which are the signatures of magnetostatic reconnection. We generalize magnetostatic reconnection to magnetic fields that vary with time, thereby including the effects of inductive electric fields. The magnetic field is, however, still specified, and is not evolved dynamically. We call this approach kinematic reconnection. The inductive electric fields are the source of variation of φ along B. The resulting field line velocity v⊥ = E x B/B2 may exhibit singularities, which are the signatures of reconnection associated with nonideal effects. Two-dimensional models (∂/∂z = 0) with X-point structures are used to illustrate this approach. Singularities of field line velocities are found on the separatrix, in addition to the singularity along the X-line. The singularities found on the separatrix agree qualitatively with the behavior seen in simulations. For a class of three-dimensional models, these singularities have distinct characteristics for models with and without nulls, suggesting different physics for the two cases. Reconnection for three-dimensional models without nulls is qualitatively the same as the two-dimensional models with a nonzero constant Bz. For models with a pair of A and B nulls, singularities appear on the Σ surfaces, the γ lines, the A-B line, and at the nulls, with fractional power-law dependences. These singularities of the field line velocities reveal the following picture of three-dimensional reconnection: field lines going into reconnection are forced to become one of the γB-BA-γA lines, and field lines coming out of reconnection are pulled from these lines. Each of the A-B lines serves as a separate bridge for the field lines to reconnect. The inductive electric field along the A-B line is the driving force of the whole process. In contrast, in the case in which no nulls occur, the inductive electric field along the closed field line-the electric field along the X-line in the two-dimensional cases - drives the reconnection.
