Change in global mean surface temperature (GMST), based on a blend of land air and ocean water temperatures, is a widely cited climate change indicator that informs the Paris Agreement goal to limit global warming since preindustrial to “well below” 2°C. Assessment of current GMST enables determination of remaining target-consistent warming and therefore a relevant remaining carbon budget. In recent IPCC reports, GMST was estimated via linear regression or differences between decade-plus period means. We propose non-linear continuous local regression (LOESS) using ±20 year windows to derive GMST across all periods of interest. Using the three observational GMST datasets with almost complete interpolated spatial coverage since the 1950s, we evaluate 1850—1900 to 2019 GMST as 1.14°C with a likely (17—83 %) range of 1.05—1.25°C, based on combined statistical and observational uncertainty, compared with linear regression of 1.05°C over 1880—2019. Performance tests in observational datasets and two model large ensembles demonstrate that LOESS, like period mean differences, is unbiased. However, LOESS also provides a statistical uncertainty estimate and gives warming through 2019, rather than the 1850—1900 to 2010—2019 period mean difference centered at the end of 2014. We derive historical global near-surface air temperature change (GSAT), using a subset of CMIP6 climate models to estimate the adjustment required to account for the difference between ocean water and ocean air temperatures. We find GSAT of 1.21°C (1.11—1.32°C) and calculate remaining carbon budgets. We argue that continuous non-linear trend estimation offers substantial advantages for assessment of long-term observational GMST.