The use of high temperature superconductors (HTSC) for cryostat current leads has been analysed for operation between liquid helium and liquid nitrogen temperatures in the conduction-cooled and vapour-cooled cases. Present fabrication methods are sufficient to produce such leads for most known applications. Thermophysical data suggest that performance of leads of HTSC operated in the normal state is nearly as good as that of pure metal leads. If leads of HTSC are optimized for the normal state and then operated in the superconducting state, helium boil-off can be reduced by about a factor of two. For leads longer than the normal state optimum, the helium boil-off can be reduced further, but the system must be guarded against thermal runaway. For temperature perturbations, the lead may enter a metastable state where a large fraction of the length remains normal. For current or magnetic field perturbations, the lead may enter a hybrid partially superconducting state in which the effective resistivity is ≈0.1 that of the normal state. In designs for these latter cases, the helium boil-off is greatly reduced below that for pure metal leads. Operation at presently obtainable current densities of 100–1000 A cm −2 results in adequate steady state cooling and thermal stability for most designs.