  In this paper, we propose and study the polar Orlicz-Minkowski problems:
under what conditions on a nonzero finite measure $\mu$ and a continuous
function $\varphi:(0,\infty)\rightarrow(0,\infty)$, there exists a convex body
$K\in\mathcal{K}_0$ such that $K$ is an optimizer of the following optimization
problems: \begin{equation*} \inf/\sup \bigg\{\int_{S^{n-1}}\varphi\big( h_L
\big) \,d \mu: L \in \mathcal{K}_{0} \ \text{and}\ |L^\circ|=\omega_{n}\bigg\}.
\end{equation*} The solvability of the polar Orlicz-Minkowski problems is
discussed under different conditions. In particular, under certain conditions
on $\varphi,$ the existence of a solution is proved for a nonzero finite
measure $\mu$ on $S^{n-1}$ which is not concentrated on any hemisphere of
$S^{n-1}.$ Another part of this paper deals with the $p$-capacitary
Orlicz-Petty bodies. In particular, the existence of the $p$-capacitary
Orlicz-Petty bodies is established and the continuity of the $p$-capacitary
Orlicz-Petty bodies is proved.
