  A detailed numerical study of magnetic reconnection in resistive MHD for very
large, previously inaccessible, Lundquist numbers ($10^4\le S\le 10^8$) is
reported. Large-aspect-ratio Sweet-Parker current sheets are shown to be
unstable to super-Alfv\'enically fast formation of plasmoid (magnetic-island)
chains. The plasmoid number scales as $S^{3/8}$ and the instability growth rate
in the linear stage as $S^{1/4}$, in agreement with the theory by Loureiro et
al. [Phys. Plasmas {\bf 14}, 100703 (2007)]. In the nonlinear regime, plasmoids
continue to grow faster than they are ejected and completely disrupt the
reconnection layer. These results suggest that high-Lundquist-number
reconnection is inherently time-dependent and hence call for a substantial
revision of the standard Sweet-Parker quasi-stationary picture for $S>10^4$.
