A 20 X 30 km2, 15 station, seismic network operated for 3 months in 1985, near Lake Bogoria in the Kenya Rift. The array provided both continuous and triggered seismic data. This thesis is concerned with the local earthquakes which occurred within 30 km of the network, in a 50 X 80 km2 study area including parts of the Rift shoulder and the central trough. 572 small events (ML < 2.7) could be located accurately (+2 km) in 3 dimensions. Unexpectedly, most of the seismic activity is associated with the major faults of the Rift shoulder, rather than the younger, minor faults in the central trough. A linear group of events in the central trough do not correspond to any surface feature, and seem to indicate a buried fault. The depth distribution of the seismicity peaks at 9 km and diminishes below 12 km, and the "brittle-ductile" transition is inferrred to occur within a 12 - 16 km depth range. This distribution is similar to those in other young intracontinental regions, suggesting a normal crustal rheology. Only a few events provided well-constrained focal mechanisms. Normal, steeply dipping, N-S striking fault plane solutions could be fitted to almost all events in the central trough. 12 of the best solutions were used to determine the stress orientation, the results indicated near-horizontal E-W extension, but this direction was poorly constrained. Suitable seismograms displayed shear wave polarisation and splitting compatible with the predictions of Extensive Dilatancy Anisotropy (EDA) theory. Instrinsic anisotropy, due to the basement fabric, is probably present, but the EDA should dominate the observations, thus allowing a determination of present-day stress orientation. Suprisingly, two dominent polarisation directions were seen in different parts of the array, indicating a change from E-W to NW-SE "tension" within the network.