Dark energy survey year 3 results: weak lensing shape catalogue

We present and characterize the galaxy shape catalogue from the first 3 yr of Dark Energy Survey (DES) observations, over an effective area of 4143 deg² of the southern sky. We describe our data analysis process and our self-calibrating shear measurement pipeline METACALIBRATION, which builds and improves upon the pipeline used in the DES Year 1 analysis in several aspects. The DES Year 3 weak-lensing shape catalogue consists of 100 204 026 galaxies, measured in the riz bands, resulting in a weighted source number density of n_eff = 5.59 gal arcmin^-2 and corresponding shape noise σ_e = 0.261. We perform a battery of internal null tests on the catalogue, including tests on systematics related to the point spread function (PSF) modelling, spurious catalogue B-mode signals, catalogue contamination, and galaxy properties.