Cyber attacks have increased in number and complexity in recent years, and companies and organizations have accordingly raised their investments in more robust infrastructure to preserve their data, assets and reputation. However, the full protection against these countless and constantly evolving threats is unattainable by the sole use of preventive measures. Therefore, to handle residual risks and contain business losses in case of an incident, firms are increasingly adopting a cyber insurance as part of their corporate risk management strategy. As a result, the cyber insurance sector-which offers to transfer the financial risks related to network and computer incidents to a third party-is rapidly growing, with recent claims that already reached a $100M dollars. However, while other insurance sectors rely on consolidated methodologies to accurately predict risks, the many peculiarities of the cyber domain resulted in carriers to often resort to qualitative approaches based on experts opinions. This paper looks at past research conducted in the area of cyber insurance and classifies previous studies in four different areas, focused respectively on studying the economical aspects, the mathematical models, the risk management methodologies, and the predictions of cyber events. We then identify, for each insurance phase, a group of practical research problems where security experts can help develop new data-driven methodologies and automated tools to replace the existing qualitative approaches.