Context: Start-up companies have become an important supplier of innovation and software-intensive products. The flexibility and reactiveness of start-ups enables fast development and launch of innovative products. However, a majority of software start-up companies fail before achieving any success. Among other factors, poor software engineering could be a significant contributor to the challenges experienced by start-ups. However, the state-of-practice of software engineering in start-ups, as well as the utilization of state-of-the-art is largely an unexplored area. Objective: In this study we investigate how software engineering is applied in start-up context with a focus to identify key knowledge areas and opportunities for further research. Method: We perform a multi-vocal exploratory study of 88 start-up experience reports. We develop a custom taxonomy to categorize the reported software engineering practices and their interrelation with business aspects, and apply qualitative data analysis to explore influences and dependencies between the knowledge areas. Results: We identify the most frequently reported software engineering (requirements engineering, software design and quality) and business aspect (vision and strategy development) knowledge areas, and illustrate their relationships. We also present a summary of how relevant software engineering knowledge areas are implemented in start-ups and identify potentially useful practices for adoption in start-ups. Conclusions: The results enable a more focused research on engineering practices in start-ups. We conclude that most engineering challenges in start-ups stem from inadequacies in requirements engineering. Many promising practices to address specific engineering challenges exists, however more research on adaptation of established practices, and validation of new start-up specific practices is needed.