Most global circulation models and climate-chemistry models forced with increasing greenhouse gases predict a strengthening of the Brewer-Dobson circulation (BDC) in the twenty-first century, and some of them claim that such strengthening has already begun at the end of the twentieth century. However, observational evidence for such a trend remains inconclusive. The goal of this paper is to examine the evidence for observed trends in the stratospheric overturning circulation using a suite of currently available observational stratospheric temperature data. Trends are examined as ''departures'' from the global mean temperature, since such trends reflect the effects of dynamics and spatially inhomogeneous radiative forcing and are to first order independent of the direct radiative effects of increasing well-mixed greenhouse gas concentrations. The primary conclusion of the study is that temperature observations do not reveal statistically significant trends in the Brewer-Dobson circulation over the period from 1979 to the present, as covered by Microwave Sounding Unit and Stratospheric Sounding Unit temperatures. The estimated trends in the BDC are weak in all datasets and not statistically significant at the 95% confidence level. In many cases, different data products yield very different results, particularly when the trends are stratified by season. Implications for the interpretation of recent stratospheric climate change are discussed. The results illustrate the essential need to better constrain the accuracy of future stratospheric temperature datasets.