Deep learning (DL) has become one of the major approaches in computational dermatopathology, evidenced by a significant increase in this topic in the current literature. We aim to provide a structured and comprehensive overview of peer-reviewed publications on DL applied to dermatopathology focused on melanoma. In comparison to well-published DL methods on non-medical images (e.g., classification on ImageNet), this field of application comprises a specific set of challenges, such as staining artifacts, large gigapixel images, and various magnification levels. Thus, we are particularly interested in the pathology-specific technical state-of-the-art. We also aim to summarize the best performances achieved thus far with respect to accuracy, along with an overview of self-reported limitations. Accordingly, we conducted a systematic literature review of peer-reviewed journal and conference articles published between 2012 and 2022 in the databases ACM Digital Library, Embase, IEEE Xplore, PubMed, and Scopus, expanded by forward and backward searches to identify 495 potentially eligible studies. After screening for relevance and quality, a total of 54 studies were included. We qualitatively summarized and analyzed these studies from technical, problem-oriented, and task-oriented perspectives. Our findings suggest that the technical aspects of DL for histopathology in melanoma can be further improved. The DL methodology was adopted later in this field, and still lacks the wider adoption of DL methods already shown to be effective for other applications. We also discuss upcoming trends toward ImageNet-based feature extraction and larger models. While DL has achieved human-competitive accuracy in routine pathological tasks, its performance on advanced tasks is still inferior to wet-lab testing (for example). Finally, we discuss the challenges impeding the translation of DL methods to clinical practice and provide insight into future research directions.