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Although most large superconducting magnets have been designed using the concept of cryostability, there is increased need for large magnets which operate at current densities above the cryostable limit (greater than 108Am-2). Large high current density superconducting magnets are chosen for the following reasons: reduced mass, reduced coil thickness or size, ana reduced cost. The design of large high current density, adiabatically stable, superconducting magnets requires a very different set of design rules than either large cryostable superconducting magnets or small self-protected high current density magnets. The problems associated with large high current density superconducting magnets fall into three categories; (a) quench protection, (b) stress and training, and (c) cryogenic design. The three categories must be considered simultaneously. The paper discusses quench protection and it implication for magnets of large stored energies (this includes strings of smaller magnets). Training and its relationship to quench protection and magnetic strain are discussed. Examples of magnets, built at the Lawrence Berkeley Laboratory and elsewhere using the design guidelines given in this report, are presented.