Abstract: Bulk (RE)-Ba-Cu-O [(RE)BCO] cuprate HTS have been developed steadily towards a wide range of sustainable engineering and technological applications since their discovery in 1986 based primarily on their unique potential to trap very large magnetic fields (>5 T) at temperatures that are accessible potentially by thermo-electric cooling techniques. Trapped fields of ∼10 T at the surface of individual (RE)BCO bulk single grains and in excess of 17 T in a reinforced two-sample stack are now being achieved reliably. This paper reviews the current state of the art of the processing of large, single grain (RE)BCO bulk superconductors required to trap fields of this magnitude, and specifically via two advanced fabrication approaches; the traditional TSMG process and the more recently developed TSIG technique. The focus of the review is on optimising the critical processing parameters to achieve high-quality, high performance single grain (RE)BCO bulk superconductors specifically for high-field applications. The review also summarises recent advances in processing, such as the integration of the so-called buffer technique into the TSMG and TSIG processing methodologies to achieve improved reliability in single grain growth with a success rate exceeding 90%, the development of a Mg-doped NdBCO generic seed crystal for the successful growth of all rare-earth and light-rare earth based bulk superconductors [(RE)BCO and (LRE)BCO] and the introduction of nano-size stable, non-superconducting phase(s) to the bulk microstructure to improve the intrinsic flux pinning strength of the material, and hence trapped magnetic field. Details of the two-step buffer-aided TSIG technique developed recently that yields dense, near-net shaped, high performance (RE)BCO bulk superconductors with improved superconducting and mechanical properties are also presented. Suitable sample-seed configurations for effective multi-seeding are discussed, which enables the production of high aspect ratio, bar-shaped (RE)BCO quasi-single grains that exhibit improved levitation forces required in Maglev-based applications, for example, are discussed. The electrical, mechanical, microstructural and magnetic properties (including those achieved from a pulsed-field magnetisation approach) of the different (RE)BCO systems are presented and the relevant correlation in properties and performance highlighted, accordingly. Finally, a brief summary of existing applications and prospects for near-future exploitation of these remarkable, technologically important materials, and particularly in the medical and pharma-industries, is provided.