The rapid development of precision medicine in recent years has started to challenge diagnostic pathology with respect to its ability to analyze histological images and increasingly large molecular profiling data in a quantitative, integrative, and standardized way. Artificial intelligence (AI) and, more precisely, deep learning technologies have recently demonstrated the potential to facilitate complex data analysis tasks, including clinical, histological, and molecular data for disease classification; tissue biomarker quantification; and clinical outcome prediction. This review provides a general introduction to AI and describes recent developments with a focus on applications in diagnostic pathology and beyond. We explain limitations including the black-box character of conventional AI and describe solutions to make machine learning decisions more transparent with so-called explainable AI. The purpose of the review is to foster a mutual understanding of both the biomedical and the AI side. To that end, in addition to providing an overview of the relevant foundations in pathology and machine learning, we present worked-through examples for a better practical understanding of what AI can achieve and how it should be done.