Platforms combining Central Processing Systems (CPUs) with Field Programmable Gate Arrays (FPGAs) have become popular, as they promise high performance with energy efficiency. This is the result of the combination of FPGA accelerators tuned to the application, with the CPU providing the programming flexibility. Unfortunately, the security of these new platforms has received little attention: The classic software security assumption that hardware is immutable no longer holds. It is expected that attack surfaces will expand and threats will evolve, hence the trust models, and security solutions should be prepared. The attacker model should be enhanced and consider the following three basic entities as the source of threats: applications run by users, accelerators designed by third-party developers, and the cloud service providers enabling the computation on their platforms. In our work, we review current trust models and existing security assumptions and point out their shortcomings. We survey existing research that target secure remote FPGA configuration, the protection of intellectual property, and secure shared use of FPGAs. When combined, these are the foundations to build a solution for secure use of FPGAs in the cloud. In addition to analysing the existing research, we provide discussions on how to improve it and disclose various concerns that have not been addressed yet.