In this paper, we present story curves, a visualization technique for exploring and communicating nonlinear narratives in movies. A nonlinear narrative is a storytelling device that portrays events of a story out of chronological order, e.g., in reverse order or going back and forth between past and future events. Many acclaimed movies employ unique narrative patterns which in turn have inspired other movies and contributed to the broader analysis of narrative patterns in movies. However, understanding and communicating nonlinear narratives is a difficult task due to complex temporal disruptions in the order of events as well as no explicit records specifying the actual temporal order of the underlying story. Story curves visualize the nonlinear narrative of a movie by showing the order in which events are told in the movie and comparing them to their actual chronological order, resulting in possibly meandering visual patterns in the curve. We also present Story Explorer, an interactive tool that visualizes a story curve together with complementary information such as characters and settings. Story Explorer further provides a script curation interface that allows users to specify the chronological order of events in movies. We used Story Explorer to analyze 10 popular nonlinear movies and describe the spectrum of narrative patterns that we discovered, including some novel patterns not previously described in the literature. Feedback from experts highlights potential use cases in screenplay writing and analysis, education and film production. A controlled user study shows that users with no expertise are able to understand visual patterns of nonlinear narratives using story curves.