We consider the linear and nonlinear evolution of disturbed magnetic X-type neutral points. The problem is formulated within a unified analytic and computational framework which highlights the essence of the magnetic annihilation process, namely, the coupling of a global convection region to a localized diffusion region surrounding the neutral point. An analytic treatment is given for the case of small disturbances of the equilibrium field in the absence of gas pressure. This problem admits well-defined azimuthal modes which allow a formally exact determination of the magnetic annihilation rate. It is shown that reconnection can only occur in the case of purely radial (m = 0) disturbances: the reconnection process is oscillatory and "fast," depending only logarithmically on the plasma resistivity (η). We show that the linear theory supports the notion of an initial implosive stage which rapidly releases the bulk of the energy associated with reconnective field disturbances. This phase is initiated by the advective focusing of the perturbation energy into the neutral point and culminates in the formation of a cylindrical diffusion region of area A ∼ η and current density J ∼ η-1. This scaling provides a signature for fast linear reconnection. Next we consider the breakdown of the linear theory. Although fast reconnection is maintained for low-amplitude disturbances in noncylindrical geometries, it is shown that finite gas pressure can stall the reconnection if sufficiently large. This effect, however, may not be critical in more complex X-point geometries. More seriously, for finite-amplitude displacements the cylindrical current structure close to the neutral point is distorted into a quasi-one-dimensional current sheet whose thickness is limited by resistive diffusion. In this case fast reconnection is consistent with a flux pileup solution in which the bulk of the energy is released as heat rather than as the kinetic energy of mass motion.