Mechanical fault diagnosis can prevent the deterioration of mechanical equipment failures and is important for the stable operation of mechanical equipment. Firstly, this paper reviews three basic methods of fault diagnosis and common methods of data-driven fault diagnosis, focusing on the characteristics and advantages of deep learning and convolutional neural networks. Then, the basic structure and working principle of CNN (Convolutional Neural Networks) and some basic methods to achieve better training results are introduced. In the next place, from data processing, data fusion, sample set construction, and so on, it is reviewed that the method of fault diagnosis based on CNN and their application scenarios and advantages and disadvantages; for another, the related knowledge and concepts of transfer learning are introduced, and some current application scenarios and advantages and disadvantages of mechanical fault diagnosis techniques combining migration learning and convolutional neural networks are reviewed. Finally, the current difficulties and challenges of convolutional neural networks are discussed, and the research directions have been prospected for CNN applied to the field of fault diagnosis. Although there is quite some similar literature reviewed, this review aims to introduce the basic methods of fault diagnosis, which draw forth the basic applications of the fault diagnosis of data-driven, CNN in the domain of fault diagnosis, and the application scenarios and advantages and disadvantages of combining TL (Transfer Learning) and CNN in fault diagnosis, as well as some problems and prospects. It helps researchers to have a basic understanding of this.