We present cosmological parameter results from the final full-mission Planck measurements of the CMB anisotropies. We find good consistency with the standard spatially-flat 6-parameter $\Lambda$CDM cosmology having a power-law spectrum of adiabatic scalar perturbations (denoted "base $\Lambda$CDM" in this paper), from polarization, temperature, and lensing, separately and in combination. A combined analysis gives dark matter density $\Omega ch2=0.120\pm 0.001$, baryon density $\Omega bh2=0.0224\pm 0.0001$, scalar spectral index $ns=0.965\pm 0.004$, and optical depth $\tau =0.054\pm 0.007$ (in this abstract we quote $68\%$ confidence regions on measured parameters and $95\%$ on upper limits). The angular acoustic scale is measured to $0.03\%$ precision, with $100\theta \ast =1.0411\pm 0.0003$. These results are only weakly dependent on the cosmological model and remain stable, with somewhat increased errors, in many commonly considered extensions. Assuming the base-$\Lambda$CDM cosmology, the inferred late-Universe parameters are: Hubble constant $H0=(67.4\pm 0.5)$km/s/Mpc; matter density parameter $\Omega m=0.315\pm 0.007$; and matter fluctuation amplitude $\sigma 8=0.811\pm 0.006$. We find no compelling evidence for extensions to the base-$\Lambda$CDM model. Combining with BAO we constrain the effective extra relativistic degrees of freedom to be $Neff=2.99\pm 0.17$, and the neutrino mass is tightly constrained to eV. The CMB spectra continue to prefer higher lensing amplitudes than predicted in base -$\Lambda$CDM at over $2\sigma$, which pulls some parameters that affect the lensing amplitude away from the base-$\Lambda$CDM model; however, this is not supported by the lensing reconstruction or (in models that also change the background geometry) BAO data. (Abridged)