The last few years have seen significant experimental progress in characterizing the copper-based hole-doped high temperature superconductors in the regime of low hole density, p. Quantum oscillations, NMR, X-ray, and STM experiments have shed much light on the nature of the ordering at low temperatures. We review evidence that the order parameter in the non-Lanthanum-based cuprates is a d-form factor density-wave. This novel order acts as an unexpected window into the electronic structure of the pseudogap phase at higher temperatures in zero field: we argue in favor of a 'fractionalized Fermi liquid' (FL*) with 4 pockets of spin S = 1/2, charge +e fermions enclosing an area specified by p. © 2016 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.