Nowadays, Virtual and Augmented Reality technologies play a supportive role in many research fields. In cultural heritage, various examples are available, including storytelling and narratives, where they can provide immersive and enhanced experiences to visitors and tourists, especially for entertainment and educational purposes. This review aims to investigate the opportunities that soundscape design and advanced sonic interactions in virtual and augmented environments can bring to cultural heritage sites and museums in terms of presence, emotional content, and cultural dissemination. Nineteen-two papers have been identified through the PRISMA methodology, and a promising positive effect of sonic interaction on user experience in a virtual environment can be observed in various studies, notwithstanding a general lack of specific contributions on the use of sound rendering and audio spatialisation for improving such experiences. Moreover, this work identifies the main involved research areas and discusses the state-of-the-art best practices and case studies where sonic interactions may assume a central role. The final part suggests possible future directions and applications for more engaging and immersive storytelling in the cultural heritage domain.