The packaging industry, which is often still based on finite resources and operates as a linear economy, needs to change in order to reduce its potential environmental impact and to become more sustainable. There are a number of ways, some of which are complementary, to actively shape the necessary transformation. The circular economy and the bioeconomy are two possibilities. In many cases, however, a combination of both approaches to a circular bioeconomy is necessary for a sustainable packaging economy in the long term. The three approaches are presented from a scientific perspective and illustrated with recent results from applied research.