This is a book about arithmetic subgroups of semisimple Lie groups, which means that we will discuss the group SL(n, Z), and certain of its subgroups. By definition, the subject matter combines algebra (groups of matrices) with number theory (properties of the integers). However, it also has important applications in geometry. In particular, arithmetic groups arise in classical differential geometry as the fundamental groups of locally symmetric spaces. (See Chapters 1 and 2 for an elaboration of this line of motivation.) They also provide important examples and test cases in geometric group theory