Psychometric functions were collected to measure biases and sensitivities in certain classical illusory configurations, such as the Müller-Lyer. We found that sensitivities (thresholds or just noticeable differences) were generally not affected by the introduction of illusory biases, and the implications of this for theories of the illusions are discussed. Experiments on the Müller-Lyer figure showed that the effect depends upon mis-location of the ends of the figure, rather than upon a global expansion as demanded by the size-constancy theory. A new illusion is described in which the perceived position of a dot is displaced towards the centre of a surrounding cluster of dots, even though it is clearly discriminable from other members of the cluster by their colour. We argue that illusions illustrate powerful constraints upon visual processing: they arise when subjects are instructed to carry out a task to which the visual system is not adapted.