I review the short-term properties of earthquake catalogs, in particular the time and size distributions and the completeness of the early part of aftershock sequences for strong, shallow earthquakes. I determine the parameters of the Omori and Gutenberg-Richter laws for aftershocks close in time to a mainshock. Aftershock sequences of large earthquakes in southern California (1952 Kern County, 1992 Joshua Tree-Landers-Big Bear sequence, 1994 Northridge, and 1999 Hector Mine), recorded in the CalTech catalog, are analyzed to demonstrate that at the beginning of these series, many small earthquakes are absent from the catalog. The number of missing earthquakes increases with the magnitude range of a catalog and for some data sets exceeds the number of aftershocks close to a mainshock listed in a catalog. Comparing global earthquake catalogs (Harvard Centroid Moment Tensor and Preliminary Determination of Epicenter) with local data sets indicates that the catalogs based on longer period waves miss many early aftershocks even when their magnitudes are well above the stated magnitude threshold. Such short-term incompleteness may introduce significant biases to the statistical analysis of the seismicity pattern, in particular for branching models of earthquake occurrence incorporating the Omori law. For such models the likelihood function strongly depends on close aftershocks. I review the techniques to alleviate this problem. Analyzing the source rupture process of several recent large earthquakes suggests that rupture propagation is highly inhomogeneous in space, time, and focal mechanism. These random variations in the rupture process can be viewed as an extension of the aftershock stochastic generating mechanism toward the origin time of a mainshock. I review various models of the earthquake rupture process and suggest that fractal distributions of microevents in time, space, and focal mechanism constitute the development of an earthquake. The final identification of an individual earthquake depends on both objective and subjective factors.