We present large-scale structure catalogues from the completed extended Baryon Oscillation Spectroscopic Survey (eBOSS). Derived from Sloan Digital Sky Survey (SDSS) IV Data Release 16 (DR16), these catalogues provide the data samples, corrected for observational systematics, and random positions sampling the survey selection function. Combined, they allow large-scale clustering measurements suitable for testing cosmological models. We describe the methods used to create these catalogues for the eBOSS DR16 Luminous Red Galaxy (LRG) and Quasar samples. The quasar catalogue contains 343 708 redshifts with 0.8 < z < 2.2 over 4808 deg(2). We combine 174 816 eBOSS LRG redshifts over 4242 deg(2) in the redshift interval 0.6 < z < 1.0 with SDSS-III BOSS LRGs in the same redshift range to produce a combined sample of 377 458 galaxy redshifts distributed over 9493 deg2. Improved algorithms for estimating redshifts allow that 98 per cent of LRG observations result in a successful redshift, with less than one per cent catastrophic failures (z > 1000 km s(-1)). For quasars, these rates are 95 and 2 per cent (with Delta z > 3000 km s(-1)). We apply corrections for trends between the number densities of our samples and the properties of the imaging and spectroscopic data. For example, the quasar catalogue obtains a chi(2)/DoF = 776/10 for a null test against imaging depth before corrections and a chi(2)/DoF= 6/8 after. The catalogues, combined with careful consideration of the details of their construction found here-in, allow companion papers to present cosmological results with negligible impact from observational systematic uncertainties.