Energy is required to heat the outer solar atmosphere to millions of degrees (refs 1, 2) and to accelerate the solar wind to hundreds of kilometres per second (refs 2-6). Alfven waves (travelling oscillations of ions and magnetic field) have been invoked as a possible mechanism to transport magneto-convective energy upwards along the Sun's magnetic field lines into the corona. Previous observations(7) of Alfvenic waves in the corona revealed amplitudes far too small (0.5 km s(-1)) to supply the energy flux (100-200 W m(-2)) required to drive the fast solar wind(8) or balance the radiative losses of the quiet corona(9). Here we report observations of the transition region (between the chromosphere and the corona) and of the corona that reveal how Alfvenic motions permeate the dynamic and finely structured outer solar atmosphere. The ubiquitous outward-propagating Alfvenic motions observed have amplitudes of the order of 20 km s(-1) and periods of the order of 100-500 s throughout the quiescent atmosphere (compatible with recent investigations(7,10)), and are energetic enough to accelerate the fast solar wind and heat the quiet corona.