As such, public-key encryption with keyword search (a.k.a PEKS or searchable encryption) does not allow the recipient to decrypt keywords i.e. encryption is not invertible. This paper introduces searchable encryption schemes which enable decryption. An additional feature is that the decryption key and the trapdoor derivation key are totally independent, thereby complying with many contexts of application. We put forward a seemingly optimal construction for decryptable searchable encryption which makes use of one KEM, one IDKEM and a couple of hash functions. We define a proper security model for decryptable searchable encryption and show that basic security requirements on the underlying KEM and IDKEM are enough for our generic construction to be strongly secure in the random oracle model.