BACKGROUND: Software engineering is a human activity. People naturally make sense of their activities and experience through storytelling. But storytelling does not appear to have been properly studied by software engineering research. AIM: We explore the question: what contribution can storytelling make to human--centric software engineering research? METHOD: We define concepts, identify types of story and their purposes, outcomes and effects, briefly review prior literature, identify several contributions and propose next steps. RESULTS: Storytelling can, amongst other contributions, contribute to data collection, data analyses, ways of knowing, research outputs, interventions in practice, and advocacy, and can integrate with evidence and arguments. Like all methods, storytelling brings risks. These risks can be managed. CONCLUSION: Storytelling provides a potential counter--balance to abstraction, and an approach to retain and honour human meaning in software engineering.