Ensuring the security and privacy of computation and data management in the cloud and edge is an ever-important requirement. There are several working solutions today for trusted computing with general purpose processors, for instance, Intel SGX and ARM TrustZone. However, with the widespread commercial adoption of specialized hardware accelerators in the cloud and at the edge, most importantly FPGAs, two questions emerge: 1) How secure are they against threats? and 2) How could FPGAs be utilized for more efficient trusted computing? In this survey, we investigate these two questions precisely. Even though there have been numerous surveys in the past on the security of FPGAs, we believe it is timely to study the space of related work again, given the large number of data-centric applications aimed at targeting trusted execution environments that have recently appeared. Therefore, in addition to presenting an overview of state of the art, we also highlight some opportunities for FPGAs in the context of providing efficient trusted computation.