The realm of textiles spans clothing, households, healthcare, sports, and industrial applications. The deformable nature of these objects poses unique challenges that prior work on rigid objects cannot fully address. The increasing interest within the community in textile perception and manipulation has led to new methods that aim to address challenges in modeling, perception, and control, resulting in significant progress. However, this progress is often tailored to one specific textile or a subcategory of these textiles. To understand what restricts these methods and hinders current approaches from generalizing to a broader range of real-world textiles, this review provides an overview of the field, focusing specifically on how and to what extent textile variations are addressed in modeling, perception, benchmarking, and manipulation of textiles. We conclude by identifying key open problems and outlining grand challenges that will drive future advancements in the field.