The answer to energy-efficient electric power transfer of high currents in the range of several tens of kA can be given by high temperature superconducting (HTS) cables. BSCCO and MgB2 have been used widely for such cables, reaching maximum currents of about 20 kA. REBCO coated conductors are promising for future HTS cables beyond 20 kA and allow the operation based on subcooled liquid nitrogen. Several cabling concepts based on REBCO tapes were developed world-wide to realize such cables. Using the stacked-Tape concept, a scalable semi-industrial process was developed by KIT, called HTS CrossConductor (HTS CroCo). Key aspects of the conceptual design of high-current HTS cables are discussed and the design of a 35 kA DC cable demonstrator made from HTS CroCo strands is presented. Aspects regarding joints, current redistribution between individual strands and electrical stabilization are highlighted. The performance of this demonstrator cable was tested, reaching the envisaged current.