  Let $G$ be a locally compact group. It is not always the case that its
reduced C*-algebra $C^*_r(G)$ admits a tracial state. We exhibit closely
related necessary and sufficient conditions for the existence of such. We gain
a complete answer when $G$ compactly generated. In particular for $G$ almost
connected, or more generally when $C^*_r(G)$ is nuclear, the existence of a
trace is equivalent to amenability. We exhibit two examples of classes of
totally disconnected groups for which $C^*_r(G)$ does not admit a tracial
state.
