This paper presents a summary of experimental findings, theoretical models and unresolved issues regarding border effects on brightness, of which the Cornsweet illusion (Cornsweet, 1970 Visual Perception. Academic Press: New York) is the best-known example. It is argued that no current theoretical model completely accounts for the wide variety of effects described. Contrast sensitivity function (CSF) models can explain many low-contrast, but not high-contrast, border effects. Lightness integration models based on Land and McCann's retinex theory (Land and McCann, 1971. J. Opt. Soc. Am. 61, pp. 1-11) have the advantage over CSF models in that they predict transitivity of border effects where they are found to occur. However, they fail to predict the appearance of a variety of Cornsweet-like figures, have never been tested with relatively high contrast versions of those figures, and have only been implemented by qualitative demonstration. It is argued that edge-detector models are potentially the most promising theoretical candidates but, as with lightness-integration models, they have invariably relied on qualitative demonstrations and have only dealt with low-contrast border effects. A computational edge-detector model which predicts the appearance of both high and low contrast Cornsweet figures is proposed and its advantages over other models, as well as its current limitations, are discussed. The final section discusses the neural locus for border effects in brightness.