  We present cosmological constraints from a cosmic shear analysis of the
fourth data release of the Kilo-Degree Survey (KiDS-1000), doubling the survey
area with nine-band optical and near-infrared photometry with respect to
previous KiDS analyses. Adopting a spatially flat $\Lambda$CDM model, we find
$S_8 = \sigma_8 (\Omega_{\rm m}/0.3)^{0.5} = 0.759^{+0.024}_{-0.021}$ for our
fiducial analysis, which is in $3\sigma$ tension with the prediction of the
Planck Legacy analysis of the cosmic microwave background. We compare our
fiducial COSEBIs (Complete Orthogonal Sets of E/B-Integrals) analysis with
complementary analyses of the two-point shear correlation function and band
power spectra, finding results to be in excellent agreement. We investigate the
sensitivity of all three statistics to a number of measurement, astrophysical,
and modelling systematics, finding our $S_8$ constraints to be robust and
dominated by statistical errors. Our cosmological analysis of different
divisions of the data pass the Bayesian internal consistency tests, with the
exception of the second tomographic bin. As this bin encompasses low redshift
galaxies, carrying insignificant levels of cosmological information, we find
that our results are unchanged by the inclusion or exclusion of this sample.
