This chapter focuses on various properties of arithmetic groups that play a central role in the arithmetic theory of algebraic groups. One of the key results, presented in sections two and three, is the construction due to Borel and Harish-Chandra of a fundamental set in the group of real points of an algebraic group defined over the rationals with respect to the subgroup of integral points. This result is then used in the fourth section to deduce a number of group-theoretic properties of arithmetic subgroups, in particular, their finite presentation. The fifth and sixth sections contain criteria for the quotient of the group of real points by the subgroup of integral points to be compact or to have finite Haar measure. The seventh section outlines some further points concerning reduction theory, including the construction of more refined fundamental sets and extensions of the previous statements to groups defined over arbitrary number fields. The eighth section discusses an open problem concerning finite arithmetic groups, while the ninth section, written for the second edition, presents results dealing with abstract arithmetic groups.