The subjectivity and inaccuracy of in-clinic Cognitive Health Assessments (CHA) have led many researchers to explore ways to automate the process to make it more objective and to facilitate the needs of the healthcare industry. Artificial Intelligence (AI) and machine learning (ML) have emerged as the most promising approaches to automate the CHA process. In this paper, we explore the background of CHA and delve into the extensive research recently undertaken in this domain to provide a comprehensive survey of the state-of-the-art. In particular, a careful selection of significant works published in the literature is reviewed to elaborate a range of enabling technologies and AI/ML techniques used for CHA, including conventional supervised and unsupervised machine learning, deep learning, reinforcement learning, natural language processing, and image processing techniques. Furthermore, we provide an overview of various means of data acquisition and the benchmark datasets. Finally, we discuss open issues and challenges in using AI and ML for CHA along with some possible solutions. In summary, this paper presents CHA tools, lists various data acquisition methods for CHA, provides technological advancements, presents the usage of AI for CHA, and open issues, challenges in the CHA domain. We hope this first-of-its-kind survey paper will significantly contribute to identifying research gaps in the complex and rapidly evolving interdisciplinary mental health field.