The spectral energy gap is an important signature that defines states of quantum matter: insulators, density waves and superconductors have very different gap structures. The momentum-resolved nature of angle-resolved photoemission spectroscopy (ARPES) makes it a powerful tool to characterize spectral gaps. ARPES has been instrumental in establishing the anisotropic d-wave structure of the superconducting gap in high-transitionerature (Tc) cuprates, which is different from the conventional isotropic s-wave superconducting gap. Shortly afterwards, ARPES demonstrated that an anomalous gap above Tc, often termed the pseudogap, follows a similar anisotropy. The nature of this poorly understood pseudogap and its relationship with superconductivity has since become the focal point of research in the field. To address this issue, the momentum, temperature, doping and materials dependence of spectral gaps have been extensively examined with significantly improved instrumentation and carefully matched experiments in recent years. This article overviews the current understanding and unresolved issues of the basic phenomenology of gap hierarchy. We show how ARPES has been sensitive to phase transitions, has distinguished between orders having distinct broken electronic symmetries, and has uncovered rich momentum- and temperature-dependent fingerprints reflecting an intertwined and competing relationship between the ordered states and superconductivity that results in multiple phenomenologically distinct ground states inside the superconducting dome. These results provide us with microscopic insights into the cuprate phase diagram. © 2014 Macmillan Publishers Limited. All rights reserved.