We give an introduction to the theory of multi-partite entanglement. We begin by describing the coordinate system of the field: Are we dealing with pure or mixed states, with single or multiple copies, what notion of locality is being used, do we aim to classify states according to their type of entanglement or to quantify it? Building on the general theory of multi-partite entanglement - to the extent that it has been achieved - we turn to explaining important classes of multi-partite entangled states, including matrix product states, stabilizer and graph states, bosonic and fermionic Gaussian states, addressing applications in condensed matter theory. We end with a brief discussion of various applications that rely on multi-partite entangled states: quantum networks, measurement-based quantum computing, non-locality, and quantum metrology.
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