The goal of this paper is to review some work on agent-based financial market models in which the dynamics is driven by piecewise-linear maps. As we will see, such models allow deep analytical insights into the functioning of financial markets, may give rise to unexpected dynamics effects, allow explaining a number of important stylized facts of financial markets, and offer novel policy recommendations. However, much remains to be done in this rather new research field. We hope that our paper attracts more scientists to this area.