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Changes in atmospheric temperature have a particular importance in climate research because climate models consistently predict a distinctive vertical profile of trends. With increasing greenhouse gas concentrations, the surface and troposphere are consistently projected to warm, with an enhancement of that warming in the tropical upper troposphere. Hence, attempts to detect this distinct ‘fingerprint’ have been a focus for observational studies. The topic acquired heightened importance following the 1990 publication of an analysis of satellite data which challenged the reality of the projected tropospheric warming. This review documents the evolution over the last four decades of understanding of tropospheric temperature trends and their likely causes. Particular focus is given to the difficulty of producing homogenized datasets, with which to derive trends, from both radiosonde and satellite observing systems, because of the many systematic changes over time. The value of multiple independent analyses is demonstrated. Paralleling developments in observational datasets, increased computer power and improved understanding of climate forcing mechanisms have led to refined estimates of temperature trends from a wide range of climate models and a better understanding of internal variability. It is concluded that there is no reasonable evidence of a fundamental disagreement between tropospheric temperature trends from models and observations when uncertainties in both are treated comprehensively.