We present the all-sky Planck catalogue of Sunyaev-Zeldovich (SZ) sources
detected from the 29 month full-mission data. The catalogue (PSZ2) is the
largest SZ-selected sample of galaxy clusters yet produced and the deepest
all-sky catalogue of galaxy clusters. It contains 1653 detections, of which
1203 are confirmed clusters with identified counterparts in external data-sets,
and is the first SZ-selected cluster survey containing > $10^3$ confirmed
clusters. We present a detailed analysis of the survey selection function in
terms of its completeness and statistical reliability, placing a lower limit of
83% on the purity. Using simulations, we find that the Y5R500 estimates are
robust to pressure-profile variation and beam systematics, but accurate
conversion to Y500 requires. the use of prior information on the cluster
extent. We describe the multi-wavelength search for counterparts in ancillary
data, which makes use of radio, microwave, infra-red, optical and X-ray
data-sets, and which places emphasis on the robustness of the counterpart
match. We discuss the physical properties of the new sample and identify a
population of low-redshift X-ray under- luminous clusters revealed by SZ
selection. These objects appear in optical and SZ surveys with consistent
properties for their mass, but are almost absent from ROSAT X-ray selected
samples.