Several slow slip events beneath the south flank of Kilauea Volcano, Hawaii, have been inferred from transient displacements in daily GPS positions. To search for smaller events that may be close to the noise level in the GPS time series, we compare displacement fields on Kilauea's south flank with displacement patterns in previously identified slow slip events. Matching displacement patterns are found for several new candidate events, although displacements are much smaller than previously identified events. One of the candidates, 29 May 2000, is coincident with a microearthquake swarm, as are all of the previously identified slow slip events. The microearthquakes follow the onset of slow slip, implying that they are triggered by stress changes during slip. The new slow slip event brings the total number of events on Kilauea, between 1997 and 2007, to eight, the smallest having M_W = 5.3, and the largest having M_W = 6.0. While the recurrence time between the four largest events is 2.11 ± 0.01 years, the repeat time for all eight events is 0.9 ± 0.6 years. We invert for the fault geometry and distribution of slip during the slow slip events. The optimal source depths of 5 km, assuming uniform slip dislocations in an elastic half‐space, are considerably shallower than the accompanying swarm earthquakes (6.5–8.5 km), which would place the earthquakes in a zone of decreased Coulomb stress. Inversions including the effects of topography and layered elastic structure in the forward models favor depths comparable to microearthquake depths, such that the earthquakes are located in a region of increased Coulomb stress. We also invert for time‐dependent fault slip directly from the 30 s GPS phase observations, constraining the source to the optimal uniform slip geometry. On the basis of these inversions, the larger events last between 1.5–2.2 days. The data are unable to resolve migration of slip along the fault. The temporal pattern of accompanying microearthquakes is consistent with the fault slip history assuming a seismicity rate theory based on rate and state‐friction, making the swarm earthquakes coshocks and aftershocks of the slow slip events.