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Many wave species have families of travelling waves - cnoidal waves and solitons - which are bounded by a wave of maximum amplitude. Remarkably, for a great many different wave systems, the limiting wave has a discontinuous slope - a so-called ''corner'' wave. Blending in previously unpublished graphs and formulas, we review both progress and unresolved difficulties in understanding corner waves. Why are they so common? What is universal about the cnoidal/corner/breaking (CCB) scenario, and what features are unique to particular wave equations? The peakons and coshoidal waves of the Camassa-Holm equation and equatorially-trapped Kelvin waves in the ocean are used as specific examples.