  We conduct a systematic study of traces on locally compact groups, in
particular traces on their universal and reduced C*-algebras. We introduce the
trace kernel, and examine its relation to the von Neumann kernel and to
small-invariant neighbourhood (SIN) quotients. In doing so, we introduce the
class of residually-$SIN$ groups, which contains both $SIN$ and maximally
almost periodic groups. We examine in detail the trace kernel for connected
groups. We study traces on reduced C*-algebras, giving a simple proof for
compactly generated groups that existence of such a trace is equivalent to
having an open normal amenable subgroup, and we display non-discrete groups
admitting unique trace.
  We finish by examining amenable traces and the factorization property. We
show for property (T) groups that amenable trace kernels coincide with von
Neumann kernels. We show for totally disconnected groups that amenable trace
separation implies the factorization property. We use amenable traces to give a
simple proof that amenability of the group is equivalent to simultaneous
nuclearity and possessing a trace of its reduced C*-algebra. As a final
application of the results obtained in the paper, we address the embeddability
of group C*-algebras into simple AF algebras. As a consequence, if a locally
compact group is amenable and tracially separated (trace kernel is trivial),
then its reduced C*-algebra is quasi-diagonal.
