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The use of generative artificial intelligence (GenAI) in academia is a subjective and hotly debated topic. Currently, there are no agreed guidelines towards the usage of GenAI systems in higher education (HE) and, thus, it is still unclear how to make effective use of the technology for teaching and learning practice. This paper provides an overview of the current state of research on GenAI for teaching and learning in HE. To this end, this study conducted a systematic review of relevant studies indexed by Scopus, using the preferred reporting items for systematic reviews and meta-analyses (PRISMA) guidelines. The search criteria revealed a total of 625 research papers, of which 355 met the final inclusion criteria. The findings from the review showed the current state and the future trends in documents, citations, document sources/authors, keywords, and co-authorship. The research gaps identified suggest that while some authors have looked at understanding the detection of AI-generated text, it may be beneficial to understand how GenAI can be incorporated into supporting the educational curriculum for assessments, teaching, and learning delivery. Furthermore, there is a need for additional interdisciplinary, multidimensional studies in HE through collaboration. This will strengthen the awareness and understanding of students, tutors, and other stakeholders, which will be instrumental in formulating guidelines, frameworks, and policies for GenAI usage.