Quantumteleportation1 provides a ‘disembodied’way to transfer quantum states fromone object to another at a distant location, assisted by previously shared entangled states and a classical communication channel. As well as being of fundamental interest, teleportation has been recognized as animportant element in long-distance quantum communication2, distributed quantumnetworks3 andmeasurementbasedquantumcomputation4,5. There have been numerous demonstrations of teleportation in different physical systems such as photons6–8, atoms9, ions10,11, electrons12 and superconducting circuits13. All the previous experiments were limited to the teleportation of one degree of freedom only. However, a single quantum particle can naturally possess various degrees of freedom—internal and external—andwith coherent coupling among them.Afundamental open challenge is to teleportmultiple degrees of freedomsimultaneously, which is necessary to describe a quantum particle fully and, therefore, to teleport it intact. Here we demonstrate quantum teleportation of the composite quantum states of a single photon encoded in both spin and orbital angularmomentum.We use photon pairs entangled in both degrees of freedom (that is, hyper-entangled) as the quantum channel for teleportation, and develop a method to project and discriminate hyper-entangledBell states by exploitingprobabilistic quantum non-demolition measurement, which can be extended to more degrees of freedom. We verify the teleportation for both spin–orbit product states and hybrid entangled states, and achieve a teleportation fidelity ranging from 0.57 to 0.68, above the classical limit. Our work is a step towards the teleportation of more complex quantum systems, and demonstrates an increase in our technical control of scalable quantum technologies.