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Recent advances in edge computing~(EC) have pushed cloud-based data caching services to edge, however, such emerging edge storage comes with numerous challenging and unique security issues. One of them is the problem of edge data integrity verification (EDIV) which coordinates multiple participants (e.g., data owners and edge nodes) to inspect whether data cached on edge is authentic. To date, various solutions have been proposed to address the EDIV problem, while there is no systematic review. Thus, we offer a comprehensive survey for the first time, aiming to show current research status, open problems, and potentially promising insights for readers to further investigate this under-explored field. Specifically, we begin by stating the significance of the EDIV problem, the integrity verification difference between data cached on cloud and edge, and three typical system models with corresponding inspection processes. To thoroughly assess prior research efforts, we synthesize a universal criteria framework that an effective verification approach should satisfy. On top of it, a schematic development timeline is developed to reveal the research advance on EDIV in a sequential manner, followed by a detailed review of the existing EDIV solutions. Finally, we highlight intriguing research challenges and possible directions for future work, along with a discussion on how forthcoming technology, e.g., machine learning and context-aware security, can augment security in EC. Given our findings, some major observations are: there is a noticeable trend to equip EDIV solutions with various functions and diversify study scenarios; completing EDIV within two types of participants (i.e., data owner and edge nodes) is garnering escalating interest among researchers; although the majority of existing methods rely on cryptography, emerging technology is being explored to handle the EDIV problem.