Melanoma, a highly prevalent and lethal form of skin cancer, has a significant impact globally. The chances of recovery for melanoma patients substantially improve with early detection. Currently, deep learning (DL) methods are gaining popularity in assisting with the early identification of melanoma. Despite their high performance, relying solely on an image classifier undermines the credibility of the application and makes it difficult to understand the rationale behind the model's predictions highlighting the need for Explainable AI (XAI). This study provides a survey on skin cancer identification using DL techniques utilized in studies from 2017 to 2024. Compared to existing survey studies, the authors address the latest related studies covering several public skin cancer image datasets and focusing on segmentation, classification based on convolutional neural networks and vision transformers, and explainability. The analysis and the comparisons of the existing studies will be beneficial for the researchers and developers in this area, to identify the suitable techniques to be used for automated skin cancer image classification. Thereby, the survey findings can be used to implement support applications advancing the skin cancer diagnosis process.