A rapidly moving crack in a brittle material is often idealized(1) as a one-dimensional object with a singular tip, moving through a two-dimensional material. However, in real three-dimensional materials, tensile cracks form a planar surface whose edge is a rapidly moving one-dimensional singular front. The dynamics of these fronts under repetitive interaction(2-4) with material inhomogeneities (asperities) and the morphology(5-11) of the fracture surface that they create are not yet understood. Here we show that perturbations(12) to a crack front in a brittle material result in long-lived and highly localized waves, which we call 'front waves'. These waves exhibit a unique characteristic shape and propagate along the crack front at approximately(13-15) the Rayleigh wave speed (the speed of sound along a free surface). Following interaction, counter-propagating front waves retain both their shape and amplitude. They create characteristic traces along the fracture surface, providing cracks with both inertia and a new mode of dissipation. Front waves are intrinsically three-dimensional, and cannot exist in conventional two-dimensional theories of fracture(1). Because front waves can transport and distribute asperity-induced energy fluctuations throughout the crack front, they may help to explain how cracks remain a single coherent entity, despite repeated interactions with randomly dispersed asperities.