In this paper we overview, compare and elaborate on the invariant representations of periodic systems. Precisely, with reference to discrete-time systems, we first introduce the concept of periodic transfer function from which a notion of generalized frequency response can be worked out. Then we discuss the following four reformulations: (i) time lifted, (ii) cyclic, (iii) frequency lifted and (iv) Fourier. A number of interesting links will be established, and many theoretical aspects somewhat overlooked in the existing literature will be clarified. All reformulations are first worked out from the input-output description and then elaborated in a state-space formalism.