The growth in polymer-based innovative packaging technology has brought about a revolution in extending the shelf life of many food and aquatic products. One of the emerging areas in this field is polymer nanocomposite (PNC) technology, which involves the incorporation of various chemicals and nanoadditives into polymers to improve their inherent properties or to add required functionality. Because the nanoparticles may interact with food components during processing, storage, or distribution and may migrate into food, PNC-based packaging materials require awareness and understanding of their potential impact on human health and the environment. Interest in migration and cytotoxic analysis of PNC has gained considerable momentum in recent years. The focus of this article is on clay-containing PNCs because the global trend in PNCs shows that 50% of all nanofillers constitute nanoclays of either natural or synthetic origin. This article presents a summary of perspectives on international regulations on test parameters and migration of chemicals from materials that come into contact with food, followed by a critical review of (1) complaints concerning the polymers, compatibilizers, and adhesive tie layers used in polymeric packages, (2) migration of constituents from PNC-based films/articles, and (3) toxicity evaluation of nanoclays and migration of nanoclays from PNCs. Finally, we believe a review article of this nature will help academic and industrial researchers who want to bring advanced PNC-based products into the market for food packaging applications.