  This paper presents a survey of results on traces and quasitraces on
C$^*$-algebras, and it provides some new results on traces on ultrapowers and
on the existence of faithful traces. As for the former, we exhibit a sequence
of traceless simple, separable, unital, nuclear C$^*$-algebras whose ultrapower
does admit a quasitrace (and likely also a trace). We characterize in different
ways C$^*$-algebras that admit a faithful trace, respectively, where each
quotient of the C$^*$-algebra admits a faithful trace.
