The physical description of all materials is rooted in quantum mechanics, which describes how atoms bond and electrons interact at a fundamental level. Although these quantum e ects can in many cases be approximated by a classical description at the macroscopic level, in recent years there has been growing interest in material systems where quantum e ects remain manifest over a wider range of energy and length scales. Such quantum materials include superconductors, graphene, topological insulators, Weyl semimetals, quantum spin liquids, and spin ices. Many of them derive their properties from reduced dimensionality, in particular from confinement of electrons to two-dimensional sheets. Moreover, they tend to be materials in which electrons cannot be considered as independent particles but interact strongly and give rise to collective excitations known as quasiparticles. In all cases, however, quantum-mechanical e ects fundamentally alter properties of the material. This Review surveys the electronic properties of quantum materials through the prism of the electron wavefunction, and examines how its entanglement and topology give rise to a rich variety of quantum states and phases; these are less classically describable than conventional ordered states also driven by quantum mechanics, such as ferromagnetism.T he way we think about manifestations of quantum physics in materials has recently undergone a profound change of perspective. Although materials scientists and engineers have long exploited quantum effects in a range of electronic deviceswell-known examples are the quantized electronic energy levels and optical selection rules at the heart of optoelectronics, and the tunnel effect that underlies the upcoming generation of harddisk drives 1,2 -the past decade has seen a dramatic increase in our understanding of how subtle quantum effects control the macroscopic behaviour of a whole range of different materials.Two strange and beautiful aspects of quantum mechanics have come to the fore. One is the topological nature of quantum wavefunctions. A familiar example is the existence of quantized vortices in superconductors. These vortices exist because of the requirement that the superconducting condensate have a well-defined phase, and gauge invariance fixes how this phase couples to magnetic flux. The phase can wind only by an integer multiple of 2π around a vortex, and this integer winding number is a simple example of a topological invariant: a quantity that remains fixed under smooth changes of a system. Similar topological quantities turn out to govern many other kinds of materials, not just superconductors, and these support phenomena ranging from dissipationless transport to novel quasiparticle excitations.Another deep feature of quantum mechanics is the non-local entanglement of some quantum states that is spectacularly highlighted in teleportation experiments with two photons separated over macroscopic distances 3 . Even the wavefunction of two spins in a singlet is entangled, in that the wavefunction of...