Open innovation (OI) means that innovation is fostered by using both external and internal influences in the innovation process. In software engineering (SE), OI has existed for decades, while we currently see a faster and broader move towards OI in SE. We therefore survey research on how OI takes place and contributes to innovation in SE. This study aims to synthesize the research knowledge on OI in the SE domain. We launched a systematic mapping study and conducted a thematic analysis of the results. Moreover, we analyzed the strength of the evidence in the light of a rigor and relevance assessment of the research. We identified 33 publications, divided into 9 themes related to OI. 17/33 studies fall in the high-rigor/high-relevance category, suggesting the results are highly industry relevant. The research indicates that start-ups have higher tendency to opt for OI compared to incumbents. The evidence also suggests that firms assimilating knowledge into their internal R&D activities, have higher likelihood of gaining financial advantages. We concluded that OI should be adopted as a complementary approach to facilitate internal innovation and not to substitute it. Further research is advised on situated OI strategies and the interplay between OI and agile practices.