We measure the clustering of quasars of the final data release (DR16) of eBOSS. The sample contains 343 708 quasars between redshifts 0.8 <= z <= 2.2 over 4699 deg2. We calculate the Legendre multipoles (0,2,4) of the anisotropic power spectrum and perform a BAO and a Full-Shape (FS) analysis at the effective redshift z(eff) = 1.480. The errors include systematic errors that amount to 1/3 of the statistical error. The systematic errors comprise a modelling part studied using a blind N-body mock challenge and observational effects studied with approximate mocks to account for various types of redshift smearing and fibre collisions. For the BAO analysis, we measure the transverse comoving distance D-M(zeff)/r(drag) = 30.60 +/- 0.90 and the Hubble distance D-H(zeff)/r(drag) = 13.34 +/- 0.60. This agrees with the configuration space analysis, and the consensus yields: D-M(zeff)/r(drag) = 30.69 +/- 0.80 and D-H(zeff)/r(drag) = 13.26 +/- 0.55. In the FS analysis, we fit the power spectrum using a model based on Regularised Perturbation Theory, which includes redshift space distortions and the Alcock-Paczynski effect. The results are D-M(zeff)/r(drag) = 30.68 +/- 0.90 and D-H(zeff)/r(drag) = 13.52 +/- 0.51 and we constrain the linear growth rate of structure f(zeff)sigma 8(z(eff)) = 0.476 +/- 0.047. Our results agree with the configuration space analysis. The consensus analysis of the eBOSS quasar sample yields: D-M(zeff)/r(drag) = 30.21 +/- 0.79, D-H(zeff)/r(drag) = 3.23 +/- 0.47, and f(z(eff))sigma(8)(z(eff)) = 0.462 +/- 0.045 and is consistent with a flat Lambda CDM cosmological model using Planck results.