We propose a new approach to characterizing the timeline of a text: temporal dependency structures, where all the events of a narrative are linked via partial ordering relations like BEFORE, AFTER, OVERLAP and IDENTITY. We annotate a corpus of children’s stories with temporal dependency trees, achieving agreement (Krippendorff’s Alpha) of 0.856 on the event words, 0.822 on the links between events, and of 0.700 on the ordering relation labels. We compare two parsing models for temporal dependency structures, and show that a deterministic non-projective dependency parser outperforms a graph-based maximum spanning tree parser, achieving labeled attachment accuracy of 0.647 and labeled tree edit distance of 0.596. Our analysis of the dependency parser errors gives some insights into future research directions.