[en] A detailed numerical study of magnetic reconnection in resistive MHD for very large, previously inaccessible, Lundquist numbers (10⁴ (le) S (le) 10⁸) is reported. Large-aspect-ratio Sweet-Parker current sheets are shown to be unstable to super-Alfvenically 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 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 quasistationary picture for S > 10⁴.