Abstract: A character on a group $G$ is a conjugation invariant function $tau$ with $tau(e) = 1$ and such that for $g_1, ldots, g_n in G$, the matrix $[tau(g_j^{-1}g_i)]$ is always non-negative definite. For finite groups, the set of extreme points in the space of characters are in one to one correspondence with the set of irreducible representations, and have been extensively studied. The study of characters on infinite groups was initiated in 1964 by Thoma who classified all characters for the group of finite permutations of $mathbb N$. In my talk I will discuss the classification of characters on certain lattices in Lie groups, generalizing results of Margulis, and present several applications related to "random subgroups", and rigidity for representations. In contrast to the combinatorial nature of Thoma's result, the techniques involved in studying characters on lattices come from ergodic theory, representation theory, and von Neumann algebras.