Excitation of localized and delocalized surface plasmon resonances can be used for turning excellent reflectors of visible light, such as gold and silver, into efficient absorbers, whose wavelength, polarization or angular bandwidths are however necessarily limited owing to the resonant nature of surface plasmon excitations involved. Nonresonant absorption has so far been achieved by using combined nano- and micro-structural surface modifications and with composite materials involving metal nanoparticles embedded in dielectric layers. Here we realize nonresonant light absorption in a well-defined geometry by using ultra-sharp convex metal grooves via adiabatic nanofocusing of gap surface plasmon modes excited by scattering off subwavelength-sized wedges. We demonstrate experimentally that two-dimensional arrays of sharp convex grooves in gold ensure efficient (>87%) broadband (450-850 nm) absorption of unpolarized light, reaching an average level of 96%. Efficient absorption of visible light by nanostructured metal surfaces open new exciting perspectives within plasmonics, especially for thermophotovoltaics.