The extreme monthly mean air and land surface temperatures simulated by the Beijing Climate Center (BCC) climate model from the sixth phase of the Coupled Model Intercomparison Project (CMIP6) are evaluated and compared with those from the version for the fifth phase Coupled Model Intercomparison Project (CMIP5) at global mean, zonal mean, and regional mean scales. Comparisons with the European Centre for Medium-Range Weather Forecasts (ECMWF) Interim Re-Analysis (ERA-Interim) and the Global Land Data Assimilation System (GLDAS) datasets indicate that both the CMIP5 and CMIP6 versions of the BCC model can simulate the warm and cold extremes of monthly air and land surface temperatures reasonably well for the period of 1979–2005. Compared with reanalyses, the CMIP6 version has better consistencies in the magnitudes, narrower spreads of zonal extremes, and smaller root mean square errors (RMSEs) than the CMIP5 version due to an update with a new cloud fraction scheme and a snow-albedo scheme. When the annual climatological cycle is removed, the CMIP6 version performs better than the CMIP5 version on the interannual variability of both warm and cold extremes. Changes in the air and land surface temperature extremes for the CMIP6 version are more similar to the reanalyses. We found that the warming rates of both warm and cold extremes at regional scales are significantly higher than the global averaged warming rate of surface temperatures, especially at the high latitudes of the Northern Hemisphere, which is consistent with the previous studies on monthly mean air temperature. Further, both the BCC models and reanalyses show that the cold extremes warm faster than the warm extremes, and the land surface temperature extremes warm more quickly than the air temperature extremes during this period.