Histopathology and genomic profiling are cornerstones of precision oncology and are routinely obtained for patients with cancer. Traditionally, histopathology slides are manually reviewed by highly trained pathologists. Genomic data, on the other hand, is evaluated by engineered computational pipelines. In both applications, the advent of modern artificial intelligence methods, specifically machine learning (ML) and deep learning (DL), have opened up a fundamentally new way of extracting actionable insights from raw data, which could augment and potentially replace some aspects of traditional evaluation workflows. In this review, we summarize current and emerging applications of DL in histopathology and genomics, including basic diagnostic as well as advanced prognostic tasks. Based on a growing body of evidence, we suggest that DL could be the groundwork for a new kind of workflow in oncology and cancer research. However, we also point out that DL models can have biases and other flaws that users in healthcare and research need to know about, and we propose ways to address them.