A character on a group is a positive definite function which takes the identity to 1 and is constant on conjugacy classes. Characters on a finite group gives an essential tool for understanding the representation theory of the group and motivated by this Thoma in 1964 initiated the study of characters on infinite groups. In 1966 Kirillov classified all characters on GL_n(k), and SL_n(k) for k a field and n \geq 2, excluding the case of SL_2(k). A number of other classification results have since been obtained for other groups by Ovcinnikov, Vershik, Kerov, and more recently by Bekka, Dudko, and Medynets, however the classification for SL_2(k) has not been completed. In my talk I will present the classification for SL_2(k) and SL_2 of some other rings and give some applications of these results. This is based on joint work with Andreas Thom.