The design and fabrication of a 32 T, 32 mm cold bore superconducting magnet with high field REBCO inner coils is underway at the NHMFL. In support of the design, conductor characterization measurements have been made including critical current as a function of field, field orientation, temperature, and strain on conductors and joints. Various conductor and turn insulation systems were examined. ... [Show full abstract] The selected coil fabrication method for the 32 T magnet is pancake wind, dry wind coils with sol-gel insulation on a stainless steel co-wind. Quench protection of the REBCO coils by distributed heaters is under development. Small REBCO coils have been made and tested in a 20 T background field to demonstrate performance of the technology. The design of the 32 T magnet is described, including coil configuration and conductor lengths, fraction of critical current, selection of conductor copper content for protection, and stress in the windings.