Compound fault, as a primary failure leading to unexpected downtime of rotating machinery, dramatically increases the difficulty in fault diagnosis. To deal with the difficulty encountered in implementing compound fault diagnosis (CFD), researchers and engineers from industry and academia have made numerous significant breakthroughs in recent years. Admittedly, many systematic surveys focused on fault diagnosis have been conducted by reputable researchers. Nevertheless, previous review articles paid more attention to fault diagnosis with several single or independent faults, resulting in that there is still lacking a comprehensive survey on CFD. Therefore, to fulfill the above requirements, it is necessary to provide an in-depth overview of fault diagnosis methods or algorithms for compound faults of rotating machinery and uncover potential challenges or opportunities that would guide and inspire readers to devote their efforts to promoting fault diagnosis technology more effective and practical. Specifically, the backgrounds including the related definitions and a new taxonomy of CFD methods are detailed according to the way of implementing compound fault recognition. Then, the state-of-the-art applications of CFD are overviewed based on relevant publications in the past decades. Finally, the challenges and opportunities associated with implementing CFD are concluded and followed by a conclusion for ending this survey. We believe that this review article can provide a systematic guideline of CFD from different aspects for potential readers and seasoned researchers.