We performed detailed photometric and astrometric analyses of the open star clusters Berkeley 68 and Stock 20. This was based on ground-based CCD UBV photometric data complemented by space-based Gaia Data Release 3 (DR3) photometry and astrometry. A total of 198 and 51 stars were identified as likely cluster members for Berkeley 68 and Stock 20, respectively. Two-color diagrams were used to derive the reddening and photometric metallicity for each cluster. The reddening for Berkeley 68 and Stock 20 is E(B - V) = 0.520 ± 0.032 mag and 0.400 ± 0.048 mag, respectively. Photometric metallicity [Fe/H] is -0.13 ± 0.08 dex for Berkeley 68 and -0.01 ± 0.06 dex for Stock 20. Keeping as constant reddening and metallicity, we determined the distance moduli and ages of the clusters through fitting isochrones to the UBV and Gaia-based color-magnitude diagrams. Photometric distances are d = 3003 ± 165 pc for Berkeley 68 and 2911 ± 216 pc for Stock 20. The cluster ages are 2.4 ± 0.2 Gyr and 50 ± 10 Myr for Berkeley 68 and Stock 20, respectively. Present-day mass function slopes were found to be Γ = 1.38 ± 0.71 and Γ = 1.53 ± 0.39 for Berkeley 68 and Stock 20, respectively. These values are compatible with the value of Salpeter. The relaxation times were estimated as 32.55 and 23.17 Myr for Berkeley 68 and Stock 20, respectively. These times are less than the estimated cluster ages, indicating that both clusters are dynamically relaxed. Orbit integration was carried out only for Berkeley 68 since radial velocity data were not available for Stock 20. Analysis indicated that Berkeley 68 was born outside the solar circle and belongs to the thin-disk component of the Milky Way.