Three-dimensional (3-D) particle simulations are performed in a double current layer configuration to investigate the stability of current sheets and boundary layers which develop during magnetic reconnection of antiparallel fields in collisionless plasma. The strong current layers that develop near the x line remain surprisingly laminar, with no evidence of turbulence and associated anomalous resistivity or viscosity. Neither the electron shear flow instabilities nor kink-like instabilities, which have been observed in these current layers in earlier simulations, are present. The sharp boundary layers which form between the inflow and outflow regions downstream of the x line are unstable to the lower hybrid drift instability. The associated fluctuations, however, do not strongly impact the rate of reconnection. As a consequence, magnetic reconnection in the 3-D system remains nearly two dimensional.