The adoption of biopolymer packaging materials to replace petroleum-based plastic packaging has become a global trend, which could reduce environmental impact and potential health threats. Therefore, the paper analyzes trends in organic packaging research and the prospects of its application. This study employs bibliometric methods to select relevant studies using a preset search string. The dynamics of publications, the most influential authors and articles, as well as the most productive institutions and countries on the topic for 2017–2022 were determined. To collect the data, Elsevier’s database Scopus was selected. The analysis revealed five salient research themes through text mining analysis: packaging and public health; packaging and children’s health safety; eco-friendly packaging and consumer behavior; food packaging and labeling; and packaging with a focus on marketing and advertising. 864 terms from 40 articles were analyzed. As a result, most selected publications focused on the impact of packaging on the environment. In addition, some publications consider it from the perspective of consumer safety. Moreover, consumers unconsciously associate organic packaging with products’ safety and quality. However, the marketing and advertising of organic packaging are insufficient, as well as the application of organic packaging in children’s food products. Acknowledgment The study was conducted within the framework of the ERASMUS+ Jean Monnet Project “Production and dissemination of circular economy ideas in accordance with the EU Action Plan.” Furthermore, Kseniia Bliumska-Danko gratefully acknowledges financial support from the Government of France within the framework of the PAUSE program (National Program of Urgent Assistance and Reception of Scientists).