High-throughput technologies such as next-generation sequencing allow biologists to observe cell function with unprecedented resolution, but the resulting datasets are too large and complicated for humans to understand without the aid of advanced statistical methods. Machine learning (ML) algorithms, which are designed to automatically find patterns in data, are well suited to this task. Yet these models are often so complex as to be opaque, leaving researchers with few clues about underlying mechanisms. Interpretable machine learning (iML) is a burgeoning subdiscipline of computational statistics devoted to making the predictions of ML models more intelligible to end users. This article is a gentle and critical introduction to iML, with an emphasis on genomic applications. I define relevant concepts, motivate leading methodologies, and provide a simple typology of existing approaches. I survey recent examples of iML in genomics, demonstrating how such techniques are increasingly integrated into research workflows. I argue that iML solutions are required to realize the promise of precision medicine. However, several open challenges remain. I examine the limitations of current state-of-the-art tools and propose a number of directions for future research. While the horizon for iML in genomics is wide and bright, continued progress requires close collaboration across disciplines.