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Due to increasing amounts of energy being supplied by intermittent renewable energy sources, future energy systems need to be able to react flexibly to sudden changes. To address this challenge, a large body of research is devoted to improving the utilization of energy flexibility of various types of energy resources on the supply and demand side. This is commonly done by applying mathematical models. The objective of this paper is to review such models and identify common modeling approaches used to quantify and optimize the use of flexible energy resources. The literature review is conducted systematically, encompassing 215 publications that feature 694 models of different energy resources. The main distinction of this work, in comparison to other reviews on the topic of energy flexibility lies in its focus on the mathematical formulation of models of energy resources and the quantification of the models' level of detail. The aim is to identify common modeling approaches and the corresponding research problems they address. The review shows that the majority of research efforts are directed towards the management and optimization of distributed energy resources. The most commonly employed modeling approach involves abstract black-box models with limited levels of detail. Linear programming and meta-heuristics are frequently used to optimize energy flexibility. A critical reflection of the current common practice is conducted and recommendations for future research are formulated. [Display omitted] • Literature review of 215 reports presenting model approaches for energy flexibility. • Categories of research problems identified and deduction of two main research focuses. • Presenting models of common energy resources classified into different model types. • Mostly low level of detail seen in modeling energy flexibility. • Optimization models are used to represent and quantify flexibility. [ABSTRACT FROM AUTHOR]