  We present cluster counts and corresponding cosmological constraints from the
Planck full mission data set. Our catalogue consists of 439 clusters detected
via 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 cosmology
sample. The counts are consistent with those from 2013 and yield compatible
constraints under the same modelling assumptions. Taking advantage of the
larger catalogue, we extend our analysis to the two-dimensional distribution in
redshift and signal-to-noise. We use mass estimates from two recent studies of
gravitational lensing of background galaxies by Planck clusters to provide
priors on the hydrostatic bias parameter, $(1-b)$. In addition, we use lensing
of cosmic microwave background (CMB) temperature fluctuations by Planck
clusters as an independent constraint on this parameter. These various
calibrations imply constraints on the present-day amplitude of matter
fluctuations in varying degrees of tension with those from the Planck analysis
of 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 it
remains substantial ($3.7\,\sigma$) for the largest estimated value. We also
examine constraints on extensions to the base flat $\Lambda$CDM model by
combining the cluster and CMB constraints. The combination appears to favour
non-minimal neutrino masses, but this possibility does little to relieve the
overall tension because it simultaneously lowers the implied value of the
Hubble parameter, thereby exacerbating the discrepancy with most current
astrophysical estimates. Improving the precision of cluster mass calibrations
from the current 10%-level to 1% would significantly strengthen these combined
analyses and provide a stringent test of the base $\Lambda$CDM model.
