Fully homomorphic encryption (FHE) has been dubbed the holy grail of cryptography, an elusive goal which could solve the IT world's problems of security and trust. Research in the area exploded after 2009 when Craig Gentry showed that FHE can be realised in principle. Since that time considerable progress has been made in finding more practical and more efficient solutions. Whilst research quickly developed, terminology and concepts became diverse and confusing so that today it can be difficult to understand what the achievements of different works actually are. The purpose of this paper is to address three fundamental questions: What is FHE? What can FHE be used for? What is the state of FHE today? As well as surveying the field, we clarify different terminology in use and prove connections between different FHE notions.