We present a joint cosmological analysis of weak gravitational lensing observations from the Kilo-Degree Survey (KiDS-1000), with redshift-space galaxy clustering observations from the Baryon Oscillation Spectroscopic Survey (BOSS) and galaxy-galaxy lensing observations from the overlap between KiDS-1000, BOSS, and the spectroscopic 2-degree Field Lensing Survey. This combination of large-scale structure probes breaks the degeneracies between cosmological parameters for individual observables, resulting in a constraint on the structure growth parameter S₈ = σ₈√(Ω_m/0.3) = 0.766_−0.014^+0.020, which has the same overall precision as that reported by the full-sky cosmic microwave background observations from Planck. The recovered S₈ amplitude is low, however, by 8.3 ± 2.6% relative to Planck. This result builds from a series of KiDS-1000 analyses where we validate our methodology with variable depth mock galaxy surveys, our lensing calibration with image simulations and null-tests, and our optical-to-near-infrared redshift calibration with multi-band mock catalogues and a spectroscopic-photometric clustering analysis. The systematic uncertainties identified by these analyses are folded through as nuisance parameters in our cosmological analysis. Inspecting the offset between the marginalised posterior distributions, we find that the S₈-difference with Planck is driven by a tension in the matter fluctuation amplitude parameter, σ₈. We quantify the level of agreement between the cosmic microwave background and our large-scale structure constraints using a series of different metrics, finding differences with a significance ranging between ∼3σ, when considering the offset in S₈, and ∼2σ, when considering the full multi-dimensional parameter space.