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). Alfvén 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 of Alfvénic waves in the corona revealed amplitudes far too small (0.5kms-1) to supply the energy flux (100-200Wm-2) required to drive the fast solar wind or balance the radiative losses of the quiet corona. Here we report observations of the transition region (between the chromosphere and the corona) and of the corona that reveal how Alfvénic motions permeate the dynamic and finely structured outer solar atmosphere. The ubiquitous outward-propagating Alfvénic motions observed have amplitudes of the order of 20kms-1 and periods of the order of 100-500s throughout the quiescent atmosphere (compatible with recent investigations), and are energetic enough to accelerate the fast solar wind and heat the quiet corona.