Abstract: Our impression of the sensory world is modified by our preceding experience. In the laboratory, this phenomenon is apparent from serial effects: participants' responses in any one trial are found to depend on previous stimuli or responses, even when task instructions and feedback establish each trial as independent. These effects are typically modelled as simple attractive or repulsive biases to single points in stimulus or response spaces. I will discuss a trio of studies which suggest instead that the influence of past experience carries rich and distinct structure; and that this structure can lead to profound and sometimes surprising effects on perception.