The growing demand for improvements in in vitro drug screening, disease modelling, and regenerative tissue therapies poses an urgent clinical need for the continued development of innovative tissue engineering strategies. Several promising biofabrication and aggregation solutions have been developed to overcome these challenges, such as three-dimensional (3D) bioprinting and chip-based physiological models. However, replicating the structural and cytoarchitectural complexities of native tissue types to achieve biomimicry of epithelial, nervous, muscular, and connective tissues remains a significant area of investigation. Innovative research efforts focused on the thread-like morphologies of bodily tissues have been developed to overcome these challenges. This review features the recent and current work done in fiber fabrication methods and the associated textile assembly techniques utilized to create fiber-based microphysiological systems, specifically for applications in disease modelling and high throughput drug testing. The advantages and disadvantages of these different fiber fabrication and assembly approaches are compared. Finally, the applications of these textile-based biofabrication approaches in epithelial, nervous, connective, and muscle tissue engineering are reviewed.