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The field of performative prediction had its beginnings in 2020 with the seminal paper "Performative Prediction" by Perdomo et al., which established a novel machine learning setup where the deployment of a predictive model causes a distribution shift in the environment, which in turn causes a mismatch between the distribution expected by the predictive model and the real distribution. This shift is defined by a so-called distribution map. In the half-decade since, a literature has emerged which has, among other things, introduced new solution concepts to the original setup, extended the setup, offered new theoretical analyses, and examined the intersection of performative prediction and other established fields. In this survey, we first lay out the performative prediction setting and explain the different optimization targets: performative stability and performative optimality. We introduce a new way of classifying different performative prediction settings, based on how much information is available about the distribution map. We survey existing implementations of distribution maps and existing methods to address the problem of performative prediction, while examining different ways to categorize them. Finally, we point out known and previously unknown connections that can be drawn to other fields, in the hopes of stimulating future research.