Scientists frequently tell stories using visualizations of scientific data, in the process of disseminating findings to peers and the general public. However, techniques and methods for effective scientific storytelling have received little attention so far. This article explores how literary and theatrical narrative conventions can inform the design and presentation of visualizations, and discusses the challenges of adapting scientific visualizations for broader audiences. It also summarizes recent workshops' findings on the role of storytelling in visualizations, and presents several examples of successful scientific-storytelling production teams. The conclusion is that scientific storytelling deserves greater support and recognition by the visualization community.