  For $\Gamma$ a cofinite Kleinian group acting on $\mathbb{H}^3$, we study the
Prime Geodesic Theorem on $M=\Gamma \backslash \mathbb{H}^3$, which asks about
the asymptotic behaviour of lengths of primitive closed geodesics (prime
geodesics) on $M$. Let $E_{\Gamma}(X)$ be the error in the counting of prime
geodesics with length at most $\log X$. For the Picard manifold,
$\Gamma=\mathrm{PSL}(2,\mathbb{Z}[i])$, we improve the classical bound of
Sarnak, $E_{\Gamma}(X)=O(X^{5/3+\epsilon})$, to
$E_{\Gamma}(X)=O(X^{13/8+\epsilon})$. In the process we obtain a mean
subconvexity estimate for the Rankin-Selberg $L$-function attached to
Maass-Hecke cusp forms. We also investigate the second moment of
$E_{\Gamma}(X)$ for a general cofinite group $\Gamma$, and show that it is
bounded by $O(X^{16/5+\epsilon})$.
