Zero-shot image recognition (ZSIR) aims at empowering models to recognize and reason in unseen domains via learning generalized knowledge from limited data in the seen domain. The gist for ZSIR is to execute element-wise representation and reasoning from the input visual space to the target semantic space, which is a bottom-up modeling paradigm inspired by the process by which humans observe the world, i.e., capturing new concepts by learning and combining the basic components or shared characteristics. In recent years, element-wise learning techniques have seen significant progress in ZSIR as well as widespread application. However, to the best of our knowledge, there remains a lack of a systematic overview of this topic. To enrich the literature and provide a sound basis for its future development, this paper presents a broad review of recent advances in element-wise ZSIR. Concretely, we first attempt to integrate the three basic ZSIR tasks of object recognition, compositional recognition, and foundation model-based open-world recognition into a unified element-wise perspective and provide a detailed taxonomy and analysis of the main research approaches. Then, we collect and summarize some key information and benchmarks, such as detailed technical implementations and common datasets. Finally, we sketch out the wide range of its related applications, discuss vital challenges, and suggest potential future directions.