Revolutionary advances in artificial intelligence (AI) in the past decade have brought transformative innovation across science and engineering disciplines. In the field of Arctic science, we have witnessed an increasing trend in the adoption of AI, especially deep learning, to support the analysis of Arctic big data and facilitate new discoveries. In this paper, we provide a comprehensive review of the applications of deep learning in sea ice remote sensing domains, focusing on problems such as sea ice lead detection, thickness estimation, sea ice concentration and extent forecasting, motion detection, and sea ice type classification. In addition to discussing these applications, we also summarize technological advances that provide customized deep learning solutions, including new loss functions and learning strategies to better understand sea ice dynamics. To promote the growth of this exciting interdisciplinary field, we further explore several research areas where the Arctic sea ice community can benefit from cutting-edge AI technology. These areas include improving multimodal deep learning capabilities, enhancing model accuracy in measuring prediction uncertainty, better leveraging AI foundation models, and deepening integration with physics-based models. We hope that this paper can serve as a cornerstone in the progress of Arctic sea ice research using AI and inspire further advances in this field.