In this paper it is proposed to describe the vertebrate remains collected by Dr. Oswald from beds of Lower Miocene age in the neighbourhood of Karungu, near the eastern shore of the Victoria Nyanza. Some other specimens of late Pliocene or Pleistocene age will be described elsewhere. The beds from which the Miocene vertebrates were collected appear to form part of the deposits of the delta of an old river, flowing into the lake at a time when its surface was considerably above its present level. These deposits are described in detail in Dr. Oswald's paper. The mammalian fauna includes: Proboscidea, especially Dinotherium hobleyi , previously described from the same locality; Artiodactyls, represented mainly by Anthracotheres ; Perissodactyls, represented by a Rhinoceros ; a small Hyracoid of peculiar character; a Rodent; and a feline Carnivore. The Reptilia are represented by Chelonians, including a new species of Cycloderma , a gigantic species of Testudo (also new), and a species of Podocnemis ; by Crocodiles of several types, including probably a species of Pristichampsa , a genus previously known only from the Eocene of France. Fishes are represented by a few vertebræ and teeth, including one of a species of Protopterus . The general character of the fauna indicates that the age of the deposits is probably Lower Miocene (Burdigalien) and that it was contemporary with the faunas of the Sables de l'Orléanais and of Moghara, and probably also with the recently-discovered fauna of the Bugti Hills in British Baluchistan. In all these localities Anthracotheres of similar