Surprising events trigger measurable brain activity and influence human behavior by affecting learning, memory, and decision-making. Currently there is, however, no consensus on the definition of surprise. Here we identify 18 mathematical definitions of surprise in a unifying framework. We first propose a technical classification of these definitions into three groups based on their dependence on an agent’s belief, show how they relate to each other, and prove under what conditions they are indistinguishable. Going beyond this technical analysis, we propose a taxonomy of surprise definitions and classify them into four conceptual categories based on the quantity they measure: (i) ‘prediction surprise’ measures a mismatch between a prediction and an observation; (ii) ‘change-point detection surprise’ measures the probability of a change in the environment; (iii) ‘confidence-corrected surprise’ explicitly accounts for the effect of confidence; and (iv) ‘information gain surprise’ measures the belief-update upon a new observation. The taxonomy poses the foundation for principled studies of the functional roles and physiological signatures of surprise in the brain.