Thirty years after it was first proposed, quantum teleportation remains one of the most important protocols in quantum information and quantum technologies, enabling the nonlocal transmission of an unknown quantum state. Quantum teleportation can be used to overcome the distance limitation in directly transferring quantum states in quantum communication and the difficulty in realizing long-range interactions among qubits in quantum computation. Since 2015, experimental quantum teleportation has moved from simple to complex quantum states (multiple degrees of freedom, high-dimensional quantum states) and from proof-of-principle demonstrations to real-world applications. We overview these advances, in particular, the understanding of the nonclassical nature of quantum teleportation, the teleportation of complex qua ntum states, progress in experiments with photons, atoms and solid-state systems and applications to quantum communication and computation, and discuss the challenges and opportunities for future developments. Sections Introduction Theoretical progress Quantum teleportation of complex quantum states Quantum teleportation for quantum communication Quantum teleportation for quantum computing Outlook 1CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, China. 2CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, China. 3Hefei National
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