The concept of public key cryptosystems (PKC) introduced by Diffie and Hellman in 1976 was a revolution in cryptography for ensuring the security of grass-root open network like Internet. Recently new revolutionary notions have come out in cryptography, which may be considered consequent upon advances of network applications like cloud computing and big data. They are functional encryption (FE) and fully homomorphic encryption (FHE). Both of them are advanced concepts of PKC. FE is advanced in the functionality of decryption from simple decryption in PKC to more sophisticated manners or decryption with computation (smart decryption). FHE is a specific class of PKC that enables anyone to compute a plaintext with keeping the encrypted form. In my talk I will briefly introduce the concept and applications of FE and FHE. I will then focus on FE for more details of the features, usages and constructions.