I will present an overview over work on the construction of Baxter's Q-operator. The latter is a an auxiliary tool in diagonalizing the Hamiltonian of integrable spin-chains. This is an alternative approach to the Bethe ansatz and has several advantages over the latter. I will highlight the representation theoretic construction of the Q-operator and discuss how this approach leads to a difference equation (called the quantum Wronskian) which is sufficient to determine the spectrum of the spin-chain Hamiltonian. In contrast to the Bethe ansatz equations which are of polynomial order (= number of sites of the chain) the quantum Wronskian leads to a system of quadratic equations. I will also briefly discuss how the Q-operator allows for an alternative description of the trace functional used in the recent discussion of correlation functions by Boos, Jimbo, Miwa, Smirnov and Takeyama.