  A countable discrete group $G$ is called Choquet-Deny if for every
non-degenerate probability measure $\mu$ on $G$ it holds that all bounded
$\mu$-harmonic functions are constant. We show that a finitely generated group
$G$ is Choquet-Deny if and only if it is virtually nilpotent. For general
countable discrete groups, we show that $G$ is Choquet-Deny if and only if none
of its quotients has the infinite conjugacy class property. Moreover, when $G$
is not Choquet-Deny, then this is witnessed by a symmetric, finite entropy,
non-degenerate measure.
