SCUBA-2 is a 10 000-bolometer submillimetre camera on the James Clerk Maxwell Telescope. The instrument commissioning was completed in 2011 September, and full science operations began in 2011 October. To harness the full potential of this powerful new astronomical tool, the instrument calibration must be accurate and well understood. To this end, the algorithms for calculating the line-of-sight opacity have been improved, and the derived atmospheric extinction relationships at both wavebands of the SCUBA-2 instrument are presented. The results from over 500 primary and secondary calibrator observations have allowed accurate determination of the flux conversion factors (FCF) for the 850 and 450 μm arrays. Descriptions of the instrument beam shape and photometry methods are presented. The calibration factors are well determined, with relative calibration accuracy better than 5 per cent at 850 μm and 10 per cent at 450 μm, reflecting the success of the derived opacity relations as well as the stability of the performance of the instrument over several months. The sample size of the calibration observations and accurate FCFs have allowed the determination of the 850 and 450 μm fluxes of several well-known submillimetre sources, and these results are compared with previous measurements from SCUBA.