  A representation $\pi$ of a locally compact group $G$ is called \e{trace
class}, if for every test function $f$ the induced operator $\pi(f)$ is a trace
class operator. The group $G$ is called \e{trace class}, if every $\pi\in G$ is
trace class. We show that trace class groups are type I and give a criterion
for semi-direct products to be trace class and show that a representation $\pi$
is trace class if and only if $\pi\otimes\pi'$ can be realized in the space of
distributions.
