Almost all processed food comes packaged in either plastic, glass, metal, or paper and paperboard materials, and many packaging materials are disposed of after a single use (linear economy). Based on the concept of a circular economy, the recycling of food packaging materials has become one of the main targets for industries and regulators around the world. However, recycling presents particular challenges, mainly related to the recycled material composition, which determines its reusability, application, functionality, and chemical safety. In this latter matter, it has been demonstrated that the use of recycled food packaging materials increases the number and possible sources of substances that could be present in the packaging material, which is of concern as substances that can migrate into food and cause health hazards upon consumption. This review compiles information regarding substances detected in non-permanent food packaging materials, focusing mainly on plastics, paper, and paperboards. The compilation of literature studies (110 research articles) on the presence of intentionally added substances (IAS) and non-intentionally added substances (NIAS) in food packaging materials, their migration, toxicity, and dietary exposure has been summarized, evaluated, and discussed. In addition, current sustainable food packaging trends have been mentioned. Finally, approaches to reduce the presence, migration, and potential exposure to substances that have migrated from packaging materials into food have been reviewed.