From a numerical study of the magnetohydrodynamic (MHD) equations we show, for the first time in three dimensions (d = 3), that velocity and magnetic-field structure functions exhibit multiscaling, extended self-similarity (ESS), and generalized extended self-similarity (GESS). We propose a new shell model for homogeneous and isotropic MHD turbulence, which preserves all the invariants of ideal MHD, reduces to a well-known shell model for fluid turbulence for zero magnetic field, has no adjustable parameters apart from Reynolds numbers, and exhibits the same multiscaling, ESS, and GESS as the MHD equations. We also study the inertial- to dissipation-range crossover. [S0031-9007(98)07096-3].