A sample of both numbered and faint Palomar-Leiden Survey (PLS) asteroids was searched for clusters in three-dimensional proper element (semimajor axis, eccentricity, sine inclination) space. These clusters, selected by stereo examination, were then filtered objectively by requiring that their Poisson probability of chance occurence be small (log P < -3.5). One-hundred four clusters were accepted as families. These families have populations from 4 to 102 members, with a median of 8 members. Well-populated families in uncrowded regions are the most certain. The Themis, Eos, and Koronis families are obvious; major families include Eunomia and Alexandra. Families are interpreted as generated by impact. The Eos and Koronis families are examples of total breakups and the Themis family is a partial breakup with all or most of the material excavated from one side of the parent body. Themis and other one-sided, partial breakups are recognizable when one large object (sometimes several) is present at one edge (in proper element space) of a family; when the proportion of ejected material is small these are more appropriately called large cratering events. Such cratering events are found with 2 Pallas, 4 Vesta, 8 Flora, 14 Irene, and 20 Massalia. Some material from Flora has been ejected into Mars crossing orbits. Structure is common in the well-populated families—the Eos and Koronis families have densely populated cores with additional material extending out from one side, Themis has a dense core attended by clumps, and Alexandra has an embedded disk. The better sampled families are discussed in terms of their geometry and taxonomy. A selection of the other families are discussed using the available proper element and physical properties data. Some families are very rich in faint PLS members and there is evidence that families do not all have the same size distribution. Many families would benefit from the discovery of additional objects and improved sampling of physical properties.