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Aims. Deep millimeter surveys are necessary to probe the dust-obscuredgalaxies at high redshift. We conducted a large observing program at 1.2 and 2mm with the NIKA2 camera installed on the IRAM 30-meter telescope. This NIKA2Cosmological Legacy Survey (N2CLS) covers two emblematic fields: GOODS-N andCOSMOS. We introduce the N2CLS survey and present new 1.2 and 2 mm number countmeasurements based on the tiered N2CLS observations from October 2017 to May2021.  Methods. We develop an end-to-end simulation that combines an input sky modelwith the instrument noise and data reduction pipeline artifacts. Thissimulation is used to compute the sample purity, flux boosting, pipelinetransfer function, completeness, and effective area of the survey. We used the117 deg$^2$ SIDES simulations as the sky model, which include the galaxyclustering. Our formalism allows us to correct the source number counts toobtain galaxy number counts, the difference between the two being due toresolution effects caused by the blending of several galaxies inside the largebeam of single-dish instruments.  Results. The N2CLS-May2021 survey reaches an average 1-$\sigma$ noise levelof 0.17 and 0.048 mJy on GOODS-N over 159 arcmin$^2$, and 0.46 and 0.14 mJy onCOSMOS over 1010 arcmin$^2$, at 1.2 and 2 mm, respectively. For a puritythreshold of 80%, we detect 120 and 67 sources in GOODS-N and 195 and 76sources in COSMOS, at 1.2 and 2 mm, respectively. Our measurement connects thebright single-dish to the deep interferometric number counts. After correctingfor resolution effects, our results reconcile the single-dish andinterferometric number counts and are further accurately compared with modelpredictions.

NIKA2 Cosmological Legacy Survey: Survey Description and Galaxy Number Counts