The major crimes of violence directed at people reviewed include assault, homicide, familial violence, rape, kidnapping, and terrorism. The major violent crimes against property that are examined include bombing, arson, and robbery. The paper finds that violence is a multidimensional rather than a unitary phenomenon. There are a variety of personality patterns associated with each of the major violent crimes, as well as a variety of motives and factors, often in conflict with one another, within each violent offender. An abstract conceptual framework termed the 'algebra of aggression' can be used to organize most of the explanatory principles that have been proposed for aggression and violence. The major constructs of the model (1) instigation to aggression, the sum of all the internal factors that motivate a person to commit a violent or aggressive act; (2) habit strength, preferences for the use of certain forms of aggressive behavior that are learned by rewarded experience; (3) inhibitions against aggression; and (4) stimulus factors, the environmental factors that facilitate or impede aggressive behavior. When the sum of the motivating factors exceeds the sum of the inhibitory factors for a given aggressive or violent act, then that act is possible. However, before that possibility becomes an actuality, it must compete with all other possible responses. Thus the response, 'competition,' is the fifth and final element in the algebra of aggression. Tables, notes, about 180 references, and an appendix presenting a list of official violent offenses are included.