Deep learning is a quite useful and proliferating technique of machine learning. Various applications, such as medical images analysis, medical images processing, text understanding, and speech recognition, have been using deep learning, and it has been providing rather promising results. Both supervised and unsupervised approaches are being used to extract and learn features as well as for the multi-level representation of pattern recognition and classification. Hence, the way of prediction, recognition, and diagnosis in various domains of healthcare including the abdomen, lung cancer, brain tumor, skeletal bone age assessment, and so on, have been transformed and improved significantly by deep learning. By considering a wide range of deep-learning applications, the main aim of this paper is to present a detailed survey on emerging research of deep-learning models for bone age assessment (e.g., segmentation, prediction, and classification). An enormous number of scientific research publications related to bone age assessment using deep learning are explored, studied, and presented in this survey. Furthermore, the emerging trends of this research domain have been analyzed and discussed. Finally, a critical discussion section on the limitations of deep-learning models has been presented. Open research challenges and future directions in this promising area have been included as well.