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It is now widely accepted that the cuprate superconductors are characterizedby the same long-range order as that present in the Bardeen-Cooper-Schrieffer(BCS) theory: that associated with the condensation of Cooper pairs. We arguethat many physical properties of the cuprates require interplay with additionalorder parameters associated with a proximate Mott insulator. We review aclassification of Mott insulators in two dimensions, and contend that theexperimental evidence so far shows that the class appropriate to the cuprateshas collinear spin correlations, bond order, and confinement of neutral, spinS=1/2 excitations. Proximity to second-order quantum phase transitionsassociated with these orders, and with the pairing order of BCS, has led tosystematic predictions for many physical properties. We use this context toreview the results of recent neutron scattering, fluxoid detection, nuclearmagnetic resonance, and scanning tunnelling microscopy experiments.Comment: 20 pages, 13 figures, non-technical review article; some technical  details in the companion review cond-mat/0211027; (v3) added refs; (v4)  numerous improvements thanks to the referees, to appear in Reviews of Modern  Physics; (v6) final version as publishe

Colloquium: Order and quantum phase transitions in the cuprate superconductors