One significant food group that is part of our daily diet is the dairy group, and both research and industry are actively involved to meet the increasing requirement for plant-based dairy alternatives (PBDAs). The production tendency of PBDAs is growing with a predictable rate of over 18.5% in 2023 from 7.4% at the moment. A multitude of sources can be used for development such as cereals, pseudocereals, legumes, nuts, and seeds to obtain food products such as vegetal milk, cheese, cream, yogurt, butter, and different sweets, such as ice cream, which have nearly similar nutritional profiles to those of animal-origin products. Increased interest in PBDAs is manifested in groups with special dietary needs (e.g., lactose intolerant individuals, pregnant women, newborns, and the elderly) or with pathologies such as metabolic syndromes, dermatological diseases, and arthritis. In spite of the vast range of production perspectives, certain industrial challenges arise during development, such as processing and preservation technologies. This paper aims at providing an overview of the currently available PBDAs based on recent studies selected from the electronic databases PubMed, Web of Science Core Collection, and Scopus. We found 148 publications regarding PBDAs in correlation with their nutritional and technological aspects, together with the implications in terms of health. Therefore, this review focuses on the relationship between plant-based alternatives for dairy products and the human diet, from the raw material to the final products, including the industrial processes and health-related concerns.