Quantum teleportation1, a way to transfer the state of a quantum system from one location to another, is central to quantum communication2 and plays an important role in a number of quantum computation protocols3,4,5. Previous experimental demonstrations have been implemented with single photonic6,7,8,9,10,11 or ionic qubits12,13. However, teleportation of single qubits is insufficient for a large-scale realization of quantum communication and computation2,3,4,5. Here, we present the experimental realization of quantum teleportation of a two-qubit composite system. In the experiment, we develop and exploit a six-photon interferometer to teleport an arbitrary polarization state of two photons. The observed teleportation fidelities for different initial states are all well beyond the state estimation limit of 0.40 for a two-qubit system14. Not only does our six-photon interferometer provide an important step towards teleportation of a complex system, it will also enable future experimental investigations on a number of fundamental quantum communication and computation protocols3,15,16,17,18.