Mixed reality technologies provide real-time and immersive experiences, which bring tremendous opportunities in entertainment, education, and enriched experiences that are not directly accessible owing to safety or cost. The research in this field has been in the spotlight in the last few years as the metaverse went viral. The recently emerging omnidirectional video streams, i.e., 360° videos, provide an affordable way to capture and present dynamic real-world scenes. In the last decade, fueled by the rapid development of artificial intelligence and computational photography technologies, the research interests in mixed reality systems using 360° videos with richer and more realistic experiences are dramatically increased to unlock the true potential of the metaverse. In this survey, we cover recent research aimed at addressing the above issues in the 360° image and video processing technologies and applications for mixed reality. The survey summarizes the contributions of the recent research and describes potential future research directions about 360° media in the field of mixed reality.