We present measurements of the cosmic microwave background (CMB) lensing potential using the final $Planck$ 2018 temperature and polarization data. We increase the significance of the detection of lensing in the polarization maps from $5\sigma$ to $9\sigma$. Combined with temperature, lensing is detected at $40\sigma$. We present an extensive set of tests of the robustness of the lensing-potential power spectrum, and construct a minimum-variance estimator likelihood over lensing multipoles $8\le L\le 400$. We find good consistency between lensing constraints and the results from the $Planck$ CMB power spectra within the $\Lambda CDM$ model. Combined with baryon density and other weak priors, the lensing analysis alone constrains $\sigma 8\Omega m0.25=0.589\pm 0.020$ ($1\sigma$ errors). Also combining with baryon acoustic oscillation (BAO) data, we find tight individual parameter constraints, $\sigma 8=0.811\pm 0.019$, $H0=67.9-1.3+1.2kms-1Mpc-1$, and $\Omega m=0.303-0.018+0.016$. Combining with $Planck$ CMB power spectrum data, we measure $\sigma 8$ to better than $1\%$ precision, finding $\sigma 8=0.811\pm 0.006$. We find consistency with the lensing results from the Dark Energy Survey, and give combined lensing-only parameter constraints that are tighter than joint results using galaxy clustering. Using $Planck$ cosmic infrared background (CIB) maps we make a combined estimate of the lensing potential over $60\%$ of the sky with considerably more small-scale signal. We demonstrate delensing of the $Planck$ power spectra, detecting a maximum removal of $40\%$ of the lensing-induced power in all spectra. The improvement in the sharpening of the acoustic peaks by including both CIB and the quadratic lensing reconstruction is detected at high significance (abridged).