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State-of-health (SOH) monitoring of lithium-ion batteries plays a key role in the reliable and safe operation of battery systems. Influenced by multiple factors, SOH is an aging path-dependent parameter, which challenges its accurate estimation and prediction. Numerous methods have been proposed for lithium-ion batteries SOH diagnostics and prognostics, but there is little discussion on how to characterize SOH. In this paper, we first review the existing characteristic parameters in defining battery SOH at cell-level and pack-level, and then propose some suggestions for SOH definitions. The impact of external influencing factors on battery degradation is introduced to lay a foundation for SOH estimation. SOH estimation methods are categorized by the parameters used to define SOH, i.e. capacity, impedance and aging-mechanism parameters. We discuss the goals of SOH monitoring and summarize its applications from short- and long-term perspectives accordingly. Finally, the challenges and future trends of battery SOH diagnostics and prognostics are thoroughly analyzed and discussed, in order to provide a comprehensive understanding for researchers and technologists. • Discuss how to comprehensively characterize battery SOH at cell-level and pack-level. • Analyze the strengths and weaknesses of the SOH estimation methods from the perspective of data acquisition sources. • Present the gaps between existing research on SOH estimations and SOH applications. • Summarize the challenges and prospects for SOH characterizations, estimations and applications. [ABSTRACT FROM AUTHOR]