The problem of current instability leading to anomalous resistivity in current sheets in which reconnection is occurring is considered. The cause of reconnection is not considered. It is simply assumed that because of external forces, oppositely directed magnetic fields are pressed together on a time scale much longer than the time scale for the development of current instability. Three regimes are delineated. (1) Current instability does not occur becuase the sheet reaches a thickness for steady state reconnection before the condition for current instability can be satisfied. (2) Current instability occurs when the sheet has contracted past the steady state thickness of the diffusion region with anomalous resistivity but before it reaches the thickness of the diffusion region in the presence of normal or inertial resistivity. This leads to a pulsating regime of reconnection in which the reconnection is rapid when anomalous resistivity becomes fully developed, the diffusion region expands so that the conditions for current instability are no longer satisfied, the anomalous resistivity decays, reconnection becomes slow, the diffusion region contracts, and the whole process repeats. (3) Current instability occurs before any part of the sheet contracts to the thickness of the diffusion region, which occupies only part of the sheet. In this case, steady state reconnection occurs with a diffusion region whose thickness is determined by anomalous resistivity. Possible applications to reconnection in the vicinity of te polar cusp, the magnetospheric tail, and solar flares are considered briefly.