Satellite-based layer average stratospheric temperature (T) climate data records (CDRs) now span more than three decades and so can elucidate climate variability associated with processes on multiple time scales. We intercompare and analyze available published T CDRs covering at least two decades, with a focus on Stratospheric Sounding Unit (SSU) and Microwave Sounding Unit (MSU) CDRs. Recent research has reduced but not eliminated discrepancies between SSU CDRs developed by NOAA and the UK Meteorological Office. The MSU CDRs from NOAA and Remote Sensing Systems are in closer agreement than the CDR from the University of Alabama in Huntsville. The latter has a previously unreported inhomogeneity in 2005, revealed by an abrupt increase in the magnitude and spatial variability of T anomaly differences between CDRs. Although time-varying biases remain in both SSU and MSU CDRs, multiple linear regression analyses reveal consistent solar, El Niño–Southern Oscillation (ENSO), quasi-biennial oscillation, aerosol, and piecewise-linear trend signals. Together, these predictors explain 80 to 90% of the variance in the near-global-average T CDRs. The most important predictor variables (in terms of percent explained variance in near-global-average T) for lower stratospheric T measured by MSU are aerosols, solar variability, and ENSO. Trends explain the largest percentage of variance in observations from all three SSU channels. In MSU and SSU CDRs, piecewise-linear trends, with a 1995 break point, indicate cooling during 1979–1994 but no trend during 1995–2013 for MSU and during 1995–2005 for SSU. These observational findings provide a basis for evaluating climate model simulations of stratospheric temperature during the past 35 years.