Circular RNAs (circRNAs) represent single-stranded RNA species that contain covalently closed 3′ and 5′ ends that provide them more stability than linear RNA, which has free ends. Emerging evidence indicates that circRNAs perform essential functions in many DNA viruses, including coronaviruses, Epstein–Barr viruses, cytomegalovirus, and Kaposi sarcoma viruses. Recent studies have confirmed that circRNAs are present in viruses, including DNA and RNA viruses, and play various important functions such as evading host immune response, disease pathogenesis, protein translation, miRNA sponges, regulating cell proliferation, and virus replication. Studies have confirmed that circRNAs can be biological signatures or pathological markers for autoimmune diseases, neurological diseases, and cancers. However, our understanding of circRNAs in DNA and RNA viruses is still limited, and functional evaluation of viral and host circRNAs is essential to completely understand their biological functions. In the present review, we describe the metabolism and cellular roles of circRNA, including its roles in various diseases and viral and cellular circRNA functions. Circular RNAs are found to interact with RNA, proteins, and DNA, and thus can modulate cellular processes, including translation, transcription, splicing, and other functions. Circular RNAs interfere with various signaling pathways and take part in vital functions in various biological, physiological, cellular, and pathophysiological processes. We also summarize recent evidence demonstrating cellular and viral circRNA’s roles in DNA and RNA viruses in this growing field of research.