This paper provides an overview of outstanding major research goals for the field of computational narrative understanding. Storytelling is an essential human practice, one that provides a sense of personal meaning, shared sense of community, and individual enjoyment. A number of research domains have increasingly focused on storytelling as a key mechanism for explaining human behavior. Now is an opportune moment to provide a vision of the contributions that computational narrative understanding can make towards this collective endeavor and the challenges facing the field. In addition to providing an overview of the elements of narrative, this paper outlines three major lines of inquiry: understanding the multimodality of narrative; the temporal patterning of narrative (narrative "shape"); and sociocultural narrative schemas, i.e. collective narratives. The paper concludes with a call for more inter-disciplinary working groups and deeper investment in building cross-cultural and multimodal narrative datasets.