Quantum plasmonics is an exciting sub-branch of nanoplasmonics where the laws of quantum theory are used to describe light-matter interactions on the nanoscale. Plasmonic materials allow extreme sub-diffraction confinement of (quantum or classical) light to regions so small that the quantization of both light and matter may be necessary for an accurate description. State of the art experiments now allow us to probe these regimes and push existing theories to the limits which opens up the possibilities of exploring the nature of many-body collective oscillations as well as developing new plasmonic devices, that use the particle quality of light and the wave quality of matter, and have a wealth of potential applications in sensing, lasing and quantum computing. This merging of fundamental condensed matter theory with application-rich electromagnetism (and a splash of quantum optics thrown in) gives rise to a fascinating area of modern physics that is still very much in its infancy. In this review we discuss and compare the keys models and experiments used to explore how the quantum nature of electrons impacts plasmonics in the context of quantum size corrections of localised plasmons and quantum tunnelling between nanoparticle dimers. We also look at some of the remarkable experiments that are revealing the quantum nature of surface plasmon polaritons.