We survey the notion of provably secure Searchable Encryption (SE) by giving a complete and comprehensive overview of the two main SE techniques: Searchable Symmetric Encryption (SSE) and Public Key Encryption with Keyword Search (PEKS). Since the pioneering work of Song, Wagner and Perrig (IEEE S&P ’00), the field of provably secure SE has expanded to the point where we felt that taking stock would provide benefit to the community. The survey has been written primarily for the non-specialist who has a basic information security background. Thus, we sacrifice full details and proofs of individual constructions in favor of an overview of the underlying key techniques. We categorize and compare the different SE schemes in terms of their security, efficiency, and functionality. For the experienced researcher we point out connections between the many approaches to SE and identify open research problems. Two major conclusions can be drawn from our work. While the so-called IND-CKA2 security notion becomes prevalent in the literature and efficient (sub-linear) SE schemes meeting this notion exist in the symmetric setting, achieving this strong form of security efficiently in the asymmetric setting remains an open problem. We observe that in multi-recipient SE schemes, regardless of their efficiency drawbacks, there is a noticeable lack of query expressiveness which hinders deployment in practice.