In recent years, there has been a significant improvement in the accuracy of the classification of pigmented skin lesions using artificial intelligence algorithms. Intelligent analysis and classification systems are significantly superior to visual diagnostic methods used by dermatologists and oncologists. However, the application of such systems in clinical practice is severely limited due to a lack of generalizability and risks of potential misclassification. Successful implementation of artificial intelligence-based tools into clinicopathological practice requires a comprehensive study of the effectiveness and performance of existing models, as well as further promising areas for potential research development. The purpose of this systematic review is to investigate and evaluate the accuracy of artificial intelligence technologies for detecting malignant forms of pigmented skin lesions. For the study, 10,589 scientific research and review articles were selected from electronic scientific publishers, of which 171 articles were included in the presented systematic review. All selected scientific articles are distributed according to the proposed neural network algorithms from machine learning to multimodal intelligent architectures and are described in the corresponding sections of the manuscript. This research aims to explore automated skin cancer recognition systems, from simple machine learning algorithms to multimodal ensemble systems based on advanced encoder-decoder models, visual transformers (ViT), and generative and spiking neural networks. In addition, as a result of the analysis, future directions of research, prospects, and potential for further development of automated neural network systems for classifying pigmented skin lesions are discussed.