Fracton phases constitute a new class of quantum state of matter. They are characterized by excitations that exhibit restricted mobility, being either immobile under local Hamiltonian dynamics or mobile only in certain directions. These phases do not wholly fit into any of the existing paradigms but connect to areas including glassy quantum dynamics, topological order, spin liquids, elasticity theory, quantum information theory, and gravity. We begin by discussing gapped fracton phases, which may be described using exactly solvable lattice spin models. We then introduce the framework of tensor gauge theory, which provides a powerful complementary perspective and allows us to access gapless fracton phases. We discuss the basic properties of gapless fracton phases and their connections to elasticity theory and gravity. We also discuss what is known about the dynamics and thermodynamics of fractons at nonzero density before concluding with a brief survey of some open problems.

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