Earthquakes cluster strongly in time and space, but it is not yet clear how much of this clustering can be explained as triggering from previous events (such as occurs for aftershock sequences following large earthquakes) and how much the clustering may reflect underlying physical processes (such as apparently drive many earthquake swarms). Triggering models such as ETAS, in which the occurrence of each earthquake increases the probability of a subsequent nearby event, can explain many properties of earthquake catalogs, including foreshock and aftershock sequences. The recent application of high-precision location methods to local earthquake catalogs permits the testing of triggering models on large numbers of small magnitude events. I will discuss three features of the southern California earthquake catalog not well explained by standard triggering models: (1) swarms and swarm-like behavior, (2) differences in precursory activity between target events of different sizes, (3) Time-symmetric time/space clustering of small earthquakes. Understanding this behavior is important because it relates to questions regarding the predictability of earthquakes, i.e., whether earthquake occurrence is part of a random process that can only be modeled statistically, or whether underlying physical processes are also involved.