We present maps, source catalogue and number counts of the largest, most complete and unbiased extragalactic submillimetre survey: the 850-μm SCUBA Half-Degree Extragalactic Survey (SHADES). Using the Submillimetre Common-User Bolometer Array (SCUBA) on the James Clerk Maxwell Telescope (JCMT), SHADES mapped two separate regions of sky: the Subaru/XMM–Newton Deep Field (SXDF) and the Lockman Hole East (LH). Encompassing 93 per cent of the overall acquired data (i.e. data taken up to 2004 February 1), these SCUBA maps cover 720 arcmin2 with a rms noise level of about 2 mJy and have uncovered >100 submillimetre galaxies. In order to ensure the utmost robustness of the resulting source catalogue, data reduction was independently carried out by four subgroups within the SHADES team, providing an unprecedented degree of reliability with respect to other SCUBA catalogues available from the literature. Individual source lists from the four groups were combined to produce a robust 120-object SHADES catalogue; an invaluable resource for follow-up campaigns aiming to study the properties of a complete and consistent sample of submillimetre galaxies. For the first time, we present deboosted flux densities for each submillimetre galaxy found in a large survey. Extensive simulations and tests were performed separately by each group in order to confirm the robustness of the source candidates and to evaluate the effects of false detections, completeness and flux density boosting. Corrections for these effects were then applied to the data to derive the submillimetre galaxy source counts. SHADES has a high enough number of detected sources that meaningful differential counts can be estimated, unlike most submillimetre surveys which have to consider integral counts. We present differential and integral source number counts and find that the differential counts are better fit with a broken power law or a Schechter function than with a single power law; the SHADES data alone significantly show that a break is required at several mJy, although the precise position of the break is not well constrained. We also find that a 850-μm survey complete down to 2 mJy would resolve 20–30 per cent of the far-infrared background into point sources.