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We present cluster counts and corresponding cosmological constraints from thePlanck full mission data set. Our catalogue consists of 439 clusters detectedvia their Sunyaev-Zeldovich (SZ) signal down to a signal-to-noise ratio of 6,and is more than a factor of 2 larger than the 2013 Planck cluster cosmologysample. The counts are consistent with those from 2013 and yield compatibleconstraints under the same modelling assumptions. Taking advantage of thelarger catalogue, we extend our analysis to the two-dimensional distribution inredshift and signal-to-noise. We use mass estimates from two recent studies ofgravitational lensing of background galaxies by Planck clusters to providepriors on the hydrostatic bias parameter, $(1-b)$. In addition, we use lensingof cosmic microwave background (CMB) temperature fluctuations by Planckclusters as an independent constraint on this parameter. These variouscalibrations imply constraints on the present-day amplitude of matterfluctuations in varying degrees of tension with those from the Planck analysisof primary fluctuations in the CMB; for the lowest estimated values of $(1-b)$the tension is mild, only a little over one standard deviation, while itremains substantial ($3.7\,\sigma$) for the largest estimated value. We alsoexamine constraints on extensions to the base flat $\Lambda$CDM model bycombining the cluster and CMB constraints. The combination appears to favournon-minimal neutrino masses, but this possibility does little to relieve theoverall tension because it simultaneously lowers the implied value of theHubble parameter, thereby exacerbating the discrepancy with most currentastrophysical estimates. Improving the precision of cluster mass calibrationsfrom the current 10%-level to 1% would significantly strengthen these combinedanalyses and provide a stringent test of the base $\Lambda$CDM model.

Planck 2015 results. XXIV. Cosmology from Sunyaev-Zeldovich cluster counts