One‐dimensional simple climate models (SCMs) play an important role within a hierarchy of climate models. They have largely been used to investigate alternative emission scenarios and estimate global‐mean temperature change. This role has expanded through the incorporation of techniques that include Monte Carlo methods and Bayesian statistics, adding the ability to generate probabilistic temperature change projections and diagnose key uncertainties, including equilibrium climate sensitivity (ECS). The latter is the most influential parameter within this class of models where it is ordinarily prescribed, rather than being an emergent property. A series of recent papers based on SCMs and Bayesian statistical methods have endeavored to estimate ECS by using instrumental observations and results from other more complex models to constrain the parameter space. Distributions for ECS depend on a variety of parameters, such as ocean diffusivity and aerosol forcing, so that conclusions cannot be drawn without reference to the joint parameter distribution. Results are affected by the treatment of natural variability, observational uncertainty, and the parameter space being explored. In addition, the highly simplified nature of SCMs means that they contain a number of implicit assumptions that do not necessarily reflect adequately the true nature of Earth's nonlinear quasi‐chaotic climate system. Differences in the best estimate and range for ECS may be partly due to variations in the structure of the SCMs reviewed in this study, along with the selection of data and the calibration details, including the choice of priors. Further investigations and model intercomparisons are needed to clarify these issues. WIREs Clim Change 2016, 7:461–473. doi: 10.1002/wcc.397