Planetary gearboxes have various merits in mechanical transmission, but their complex structure and intricate operation modes bring large challenges in terms of fault diagnosis. Deep learning has attracted increasing attention in intelligent fault diagnosis and has been successfully adopted for planetary gearbox fault diagnosis, avoiding the difficulty in manually analyzing complex fault features with signal processing methods. This paper presents a comprehensive review of deep learning-based planetary gearbox health state recognition. First, the challenges caused by the complex vibration characteristics of planetary gearboxes in fault diagnosis are analyzed. Second, according to the popularity of deep learning in planetary gearbox fault diagnosis, we briefly introduce six mainstream algorithms, i.e., autoencoder (AE), deep Boltzmann machine (DBM), convolutional neural network (CNN), transformer, generative adversarial network (GAN), and graph neural network (GNN), and some variants of them. Then, the applications of these methods to planetary gearbox fault diagnosis are reviewed. Finally, the research prospects and challenges in this research are discussed. According to the challenges, a dataset is introduced in this paper to facilitate future investigations. We expect that this paper can provide new graduate students, institutions and companies with a preliminary understanding of methods used in this field. The dataset can be downloaded from https://github.com/Liudd-BJUT/WT-planetary-gearbox-dataset.