There is a strong drive in industry for packaging solutions that contribute to sustainable development by targeting a circular economy, which pivots around the recyclability of the packaging materials. The aim is to reduce traditional plastic consumption and achieve high recycling efficiency while maintaining the desired barrier and mechanical properties. In this domain, packaging materials in the form of polymer nanocomposites (PNCs) can offer the desired functionalities and can be a potential replacement for complex multilayered polymer structures. There has been an increasing interest in nanocomposites for food packaging applications, with a five-fold rise in the number of published articles during the period 2010–2019. The barrier, mechanical, and thermal properties of the polymers can be significantly improved by incorporating low concentrations of nanofillers. Furthermore, antimicrobial and antioxidant properties can be introduced, which are very relevant for food packaging applications. In this review, we will present an overview of the nanocomposite materials for food packaging applications. We will briefly discuss different nanofillers, methods to incorporate them in the polymer matrix, and surface treatments, with a special focus on the barrier, antimicrobial, and antioxidant properties. On the practical side migration issues, consumer acceptability, recyclability, and toxicity aspects will also be discussed.