<p>Early fall detection among the elderly can dramatically reduce the risk of major problems, including disabling fractures and psychological issues, with falls potentially leading to severe consequences, including death. Artificial Intelligence has emerged as a valuable tool, leading to the development of various automatic fall detection systems for elderly people. The main objective of this paper is to analyze the state of the art in Elderly Fall Detection Systems while maintaining a focus on the explainability of results. A comprehensive analysis of the literature will be conducted, examining works that employ both standard Artificial Intelligence techniques and eXplainable Artificial Intelligence (XAI) approaches. This work aims to provide a solid foundation for researchers in this domain by offering a multi-level analysis: examining proposed solutions (both explainable and non-explainable), analyzing various data types utilized, and exploring relevant medical aspects closely related to the domain. The literature review follows standard PRISMA guidelines for systematic reviews. Major academic platforms were leveraged to extract relevant papers and generate the dataset. We categorized Elderly Fall Detection System applications based on different explainable and non-explainable artificial intelligence techniques. Various evaluation metrics were established to assess the techniques used for prediction, detection, recognition, classification, and identification tasks. The paper concludes with an in-depth discussion of current developments and potential future directions, emphasizing the importance of trustworthy artificial intelligence systems that combine high operating accuracy, transparency, rapid response times, and reliability.</p>