To use quantum systems for technological applications we first need to preserve their coherence for macroscopic timescales, even at finite temperature. Quantum error correction has made it possible to actively correct errors that affect a quantum memory. An attractive scenario is the construction of passive storage of quantum information with minimal active support. Indeed, passive protection is the basis of robust and scalable classical technology, physically realized in the form of the transistor and the ferromagnetic hard disk. The discovery of an analogous quantum system is a challenging open problem, plagued with a variety of no-go theorems. Several approaches have been devised to overcome these theorems by taking advantage of their loopholes. Here we review the state-of-the-art developments in this field in an informative and pedagogical way. We give the main principles of self-correcting quantum memories and we analyze several milestone examples from the literature of two-, three- and higher-dimensional quantum memories.

v2,3,4 - Final author copy; to appear in Rev. Mod. Phys.; 55 pages, 29 figures, 254 references; improvements include new sections on the Curie-Weiss model, open questions on interacting anyon models, and a discussion on SPT phases. Criteria for self correction is simplified, and the discussion on the 4D toric code is extended. Other changes are made following new work since submission of v1