Historical changes in global atmospheric temperature are typically estimated using simple linear trends. This paper considers three alternative simple statistical models, each involving breakpoints (abrupt changes): a flat steps model, in which all changes occur abruptly; a piecewise linear model; and a sloped steps model, incorporating both abrupt changes and slopes during the periods between breakpoints. First- and second-order autoregressive models are used in combination with each of the above. Goodness of fit of the models is evaluated using the Schwarz Bayesian Information Criterion. These models are applied to the instrumental record of global monthly temperature anomalies at the surface and to the radiosonde and satellite records for the troposphere and stratosphere. The alternative models often provide a better fit to the observations than the simple linear model. Typically the two top-performing models have very close values of the Schwarz Bayesian Information Criterion. Usually the two models have the same basic form and the same net temperature change but with a different choice of autoregressive model. However, in some cases the best fits are from two different basic models, yielding different net temperature changes and suggesting different interpretations of the nature of those changes. For the surface data during 1900--2002 the sloped steps and piecewise linear models offer the best fits. Results for tropospheric data suggest that it is reasonable to consider most of the warming during 1958--2001 to have occurred at the time of the abrupt climate regime shift in 1977. Two fundamentally different, but equally valid, descriptions of stratospheric cooling were found: gradual linear change versus more abrupt ratcheting down of temperature concentrated in postvolcanic periods (~2 years after eruption). Because models incorporating abrupt changes can be as explanatory as simple linear trends, we suggest consideration of these alternatives in climate change detection and attribution studies.