In flows where the ratio of inertia to gravity varies strongly, large variations in the fluid thickness appear and hydraulic jumps arise, as depicted by Rayleigh. We report a new family of hydraulic jumps, where the capillary effects dominate the gravitational acceleration. The Bond number – which measures the importance of gravitational body forces compared to surface tension – must be small in order to observe such objects using capillarity as a driving force. For water, the typical length should be smaller than 3 mm. Nevertheless, for such small scales, solid boundaries induce viscous stresses, which dominate inertia, and capillary jumps should not be described by the inertial shock wave theory that one would deduce from Bélanger or Rayleigh for hydraulic jumps. In order to get rid of viscous shears, we consider Plateau borders, which are the microchannels defined by the merging of three films inside liquid foams, and we show that capillary jumps propagate along these deformable conduits. We derive a simple model that predicts the velocity, geometry and shape of such fronts. A strong analogy with Rayleigh’s description is pointed out. In addition, we carried out experiments on a single Plateau border generated with soap films to observe and characterize these capillary jumps. Our theoretical predictions agree remarkably well with the experimental measurements.