In this talk, I will give an overview of results and open problems in the study of quantum homomorphic encryption, the quantum version of the classical cryptographic primitive: fully homomorphic encryption. Classical fully homomorphic encryption allows data to be encrypted in such a way that a third party, without the ability to decrypt, can perform computations on the encrypted data. In quantum homomorphic encryption, we wish to extend these results to encrypting *quantum data* and to applying *quantum circuits* to the encrypted data.

This talk is meant to be accessible to a broad audience. No expertise in quantum computing or classical cryptography will be assumed.