We present the first results from the largest deep extragalactic mm-wavelength survey undertaken to date. These results are derived from maps covering over 0.7 deg2, made at λ= 1.1 mm, using the AzTEC continuum camera mounted on the James Clerk Maxwell Telescope. The maps were made in the two fields originally targeted at λ= 850 μm with the Submillimetre Common-User Bolometer Array (SCUBA) in the SCUBA Half-Degree Extragalactic Survey (SHADES) project, namely the Lockman Hole East (mapped to a depth of 0.9–1.3 mJy rms) and the Subaru/XMM–Newton Deep Field (mapped to a depth of 1.0–1.7 mJy rms). The wealth of existing and forthcoming deep multifrequency data in these two fields will allow the bright mm source population revealed by these new wide-area 1.1 mm images to be explored in detail in subsequent papers. Here, we present the maps themselves, a catalogue of 114 high-significance submillimetre galaxy detections, and a thorough statistical analysis leading to the most robust determination to date of the 1.1 mm source number counts. These new maps, covering an area nearly three times greater than the SCUBA SHADES maps, currently provide the largest sample of cosmological volumes of the high-redshift Universe in the mm or sub-mm. Through careful comparison, we find that both the Cosmic Evolution Survey (COSMOS) and the Great Observatories Origins Deep Survey (GOODS) North fields, also imaged with AzTEC, contain an excess of mm sources over the new 1.1 mm source-count baseline established here. In particular, our new AzTEC/SHADES results indicate that very luminous high-redshift dust enshrouded starbursts (S1.1mm > 3 mJy) are 25–50 per cent less common than would have been inferred from these smaller surveys, thus highlighting the potential roles of cosmic variance and clustering in such measurements. We compare number count predictions from recent models of the evolving mm/sub-mm source population to these sub-mm bright galaxy surveys, which provide important constraints for the ongoing refinement of semi-analytic and hydrodynamical models of galaxy formation, and find that all available models overpredict the number of bright submillimetre galaxies found in this survey.