Primary cosmic-ray particles, detected by means of the extensive cascades they generate in the atmosphere, have been observed over a continuous range of energies up to 1020 eV, and apparently somewhat higher. At energies such that the radius of curvature of their trajectories, if they are protons, as expected, is comparable to our distance from the galactic centre, the arrival directions of 84 observed particles are distributed randomly over the sky. The energy spectrum of the particles shows an anomaly near 1015 eV, where the flux is higher than expected by extrapolation of data near 1012 eV, and then falls very rapidly (spectral exponent y » 3.5 at energies just above 4 x 1015 eV). Above 1017 eV the flux falls off less rapidly, y being near 3.0 in the range 1018 to 3 x 1019 eV. Extrapolating the flux back to low energies from 1018 eV, where the particles are often assumed to be of extra-galactic origin, gives a flux higher than that actually observed at low energies. The best evidence on energies of the large showers indicates that these are above 1020 eV, which is greater than the upper limit to which metagalactic protons could survive interactions with microwave photons. There is evidence that many of the most energetic particles (near 1018 eV) are indeed protons, but this result is only preliminary.