Skin cancer is one of the most dangerous diseases in the world. Correctly classifying skin lesions at an early stage could aid clinical decision-making by providing an accurate disease diagnosis, potentially increasing the chances of cure before cancer spreads. However, achieving automatic skin cancer classification is difficult because the majority of skin disease images used for training are imbalanced and in short supply; meanwhile, the model’s cross-domain adaptability and robustness are also critical challenges. Recently, many deep learning-based methods have been widely used in skin cancer classification to solve the above issues and achieve satisfactory results. Nonetheless, reviews that include the abovementioned frontier problems in skin cancer classification are still scarce. Therefore, in this article, we provide a comprehensive overview of the latest deep learning-based algorithms for skin cancer classification. We begin with an overview of three types of dermatological images, followed by a list of publicly available datasets relating to skin cancers. After that, we review the successful applications of typical convolutional neural networks for skin cancer classification. As a highlight of this paper, we next summarize several frontier problems, including data imbalance, data limitation, domain adaptation, model robustness, and model efficiency, followed by corresponding solutions in the skin cancer classification task. Finally, by summarizing different deep learning-based methods to solve the frontier challenges in skin cancer classification, we can conclude that the general development direction of these approaches is structured, lightweight, and multimodal. Besides, for readers’ convenience, we have summarized our findings in figures and tables. Considering the growing popularity of deep learning, there are still many issues to overcome as well as chances to pursue in the future.