A polarimeter has been built for use with the Submillimetre Common-User Bolometer Array (SCUBA), on the James Clerk Maxwell Telescope (JCMT) in Hawaii. SCUBA is the first of a new generation of highly sensitive submillimetre cameras, and the UK/Japan Polarimeter adds a polarimetric imaging/photometry capability in the wavelength range 350 to 2000 microns. Early science results range from measuring the synchrotron polarization of the black hole candidate Sgr A* to mapping magnetic fields inferred from polarized dust emission in Galactic star-forming clouds. We describe the instrument design, performance, observing techniques and data reduction processes, along with an assessment of the current and future scientific capability.