A trace on a group is a positive-definite conjugation-invariant function on it. These functions play an important role in harmonic analysis of discrete groups, and their study has found many exciting connections to rigidity, stability, and dynamics in the past couple of decades. In this talk, I will explain these connections and focus on the topological structure of the space of traces of some groups. We will then see the different behaviours of these spaces for free groups vs. higher-rank lattices. Finally, some open questions about free products and surface groups will be presented. This is based on joint works with Arie Levit, Joav Orovitz and Itamar Vigdorovich.