Identifications of radio sources with galaxies and clusters of galaxies have been sought systematically in a limited region of the sky. The optical data have been taken principally from the National Geographic Society-Palomar Observatory Sky Atlas and a catalogue of clusters of galaxies prepared by Abell (1958) from the same Atlas. The radio data are taken from results obtained with the Sydney cross-type radio telescope, supplemented at times with additional information from a recent Cambridge Catalogue (3C). A total of 46 possible identifications with galaxies are listed and 55 possible identifications with clusters of galaxies, the great majority of which are new. Most of these galaxies are double systems, but no other common features could be recognized: it seems probable that many galaxies of completely normal appearance are very strong radio emitters. The possible nature of the double galaxies is discussed briefly and attention is drawn to a corresponding duplicity in the radio brightness distribution observed in some strong radio sources. In many cases the emission from clusters appears to be associated with a single galaxy or pair of galaxies in the cluster; evidence for the existence of radiation of intergalactic origin is inconclusive. A selection of the possible galaxy identifications has been used to derive a provisional radio luminosity function; it appears probable that there is no distinct class of " radio galaxies", but all radio luminosities appear to be represented, the numbers of radio sources in a given volume of space falling rapidly with increasing luminosity. A radio luminosity function of the form derived is capable of accounting for a substantial number of the radio sources of small size observed at high galactic latitudes; the remainder could be accommodated by the uncertainty in the data. It is found that, at the levels to which source counts can currently be taken, cosmological effects are likely to be small, although the most distant sources are markedly affected.