We describe the POLAMI program for the monitoring of all four Stokes parameters of a sample of bright radio-loud active galactic nuclei with the IRAM 30m telescope at 3.5 and 1.3mm. The program started in October 2006 and accumulated, until August 2014, 2300 observations at 3.5mm, achieving a median time sampling interval of 22 days for the sample of 37 sources. This first paper explains the source selection, mostly blazars, the observing strategy and data calibration, and gives the details of the instrumental polarisation corrections. The sensitivity (1sigma) reached at 3.5mm is 0.5% (linear polarisation degree), 4.7 deg. (polarisation angle), and 0.23% (circular polarisation), while the corresponding values at 1.3mm are 1.7%, 9.9 deg., and 0.72%, respectively. The data quality is demonstrated by the time sequences of our calibrators Mars and Uranus. For the quasar 3C286, widely used as a linear polarisation calibrator, we give improved estimates of its linear polarisation, and show for the first time occasional detections of its weak circular polarisation, which suggests a small level of variability of the source at millimeter wavelengths.
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