  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.
