Biological systems are influenced by fluid mechanics at nearly all spatiotemporal scales. This broad relevance of fluid mechanics to biology has been increasingly appreciated by engineers and biologists alike, leading to continued expansion of research in the field of biological fluid dynamics. While this growth is exciting, it can present a barrier to researchers seeking a concise introduction to key challenges and opportunities for progress in the field. Rather than attempt a comprehensive review of the literature, this article highlights a limited selection of classic and recent work. In addition to motivating the study of biological fluid dynamics in general, the goal is to identify both longstanding and emerging conceptual questions th...