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Wide, deep, blind continuum surveys at submillimetre/millimetre (submm/mm) wavelengths are required to provide a full inventory of the dusty, distant Universe. However, conducting such surveys to the necessary depth, with sub-arcsec angular resolution, is prohibitively time-consuming, even for the most advanced submm/mm telescopes. Here, we report the most recent results from the ALMACAL project, which exploits the 'free' calibration data from the Atacama Large Millimetre/submillimetre Array (ALMA) to map the lines of sight towards and beyond the ALMA calibrators. ALMACAL has now covered 1,001 calibrators, with a total sky coverage around 0.3 deg2, distributed across the sky accessible from the Atacama desert, and has accumulated more than 1,000h of integration. The depth reached by combining multiple visits to each field makes ALMACAL capable of searching for faint, dusty, star-forming galaxies (DSFGs), with detections at multiple frequencies to constrain the emission mechanism. Based on the most up-to-date ALMACAL database, we report the detection of 186 DSFGs with flux densities down to S870um ~ 0.2mJy, comparable with existing ALMA large surveys but less susceptible to cosmic variance. We report the number counts at five wavelengths between 870um and 3mm, in ALMA bands 3, 4, 5, 6 and 7, providing a benchmark for models of galaxy formation and evolution. By integrating the observed number counts and the best-fitting functions, we also present the resolved fraction of the cosmic infrared background (CIB) and the CIB spectral shape. Combining existing surveys, ALMA has currently resolved about half of the CIB in the submm/mm regime.