We performed detailed photometric and astrometric analyses of the open star clusters Berkeley68 and Stock20. 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 Berkeley68 and Stock20, respectively. Two-color diagrams were used to derive the reddening and photometric metallicity for each cluster. The reddening for Berkeley68 and Stock20 is E(B-V)=0.520{+/-}0.032mag and 0.400{+/-}0.048mag, respectively. Photometric metallicity [Fe/H] is -0.13{+/-}0.08dex for Berkeley68 and -0.01{+/-}0.06dex for Stock20. 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{+/-}165pc for Berkeley68 and 2911{+/-}216pc for Stock20. The cluster ages are 2.4{+/-}0.2Gyr and 50{+/-}10Myr for Berkeley68 and Stock20, respectively. Present-day mass function slopes were found to be {Gamma}=1.38{+/-}0.71 and {Gamma}=1.53{+/-}0.39 for Berkeley68 and Stock20, respectively. These values are compatible with the value of Salpeter. The relaxation times were estimated as 32.55 and 23.17Myr for Berkeley68 and Stock20, respectively. These times are less than the estimated cluster ages, indicating that both clusters are dynamically relaxed. Orbit integration was carried out only for Berkeley68 since radial velocity data were not available for Stock20. Analysis indicated that Berkeley68 was born outside the solar circle and belongs to the thin-disk component of the Milky Way.