For what algebraic systems does a useful privacy homomorphism exist?

Homomorphic encryption plays a crucial role in the challenging problem of privacy preservation. In this survey, we describe a number of homomorphic schemes providing the relevant definitions to make the topic accessible to both cryptographers and mathematicians. We classify the schemes according to the timeline of appearance and, for some of them, we verify that they are correct with respect to decryption and evaluation, providing proofs or references. Recent research directions are also briefly discussed in this context.