Let $H^0(X)$ ($H(X)$) denote the set of all (non-empty) closed subsets of $X$ endowed with the Vietoris topology. A basic problem concerning $H(X)$ is to characterize those $X$ for which $H(X)$ is countably compact. We conjecture that $u$-compactness of $X$ for some $u \in \omega^*$ (or equivalently: all powers of $X$ are countably compact) may be such a characterization. We give some results that point into this direction.

We define the property $R(\kappa)$: for every family $\{Z_\alpha:\alpha<\kappa\}$ of closed subsets of $X$ separated by pairwise disjoint open sets and any family $\{ k_\alpha : \alpha < \kappa \}$ of natural numbers, the product $\Pi_{\alpha<\kappa}Z_\alpha^{k_\alpha}$ is countably compact, and prove that if $H(X)$ is countably compact for a $T_2$-space $X$ then $X$ satisfies $R(\kappa$) for all $\kappa$. A space has $R(1)$ iff all its finite powers are countably compact, so this generalizes a theorem of J. Ginsburg : if $X$ is $T_2$ and $H(X$) is countably compact, then so is $X^n$ for all $n<\omega$. We also prove that, for $\kappa < \mathfrak{t}$, if the $T_3$ space $X$ satisfies a weak form of $R(\kappa$), the orbit of every point in $X$ is dense, and $X$ contains $\kappa$ pairwise disjoint open sets, then $X^\kappa$ is countably compact. This generalizes the following theorem of J. Cao, T. Nogura, and A. Tomita : if $X$ is $T_3$, homogeneous, and $H(X)$ is countably compact, then so is $X^\omega$.

Then we study the Frol\'{\i}k sum (also called ``one-point countable-compactification") $F(X_\alpha:\alpha<\kappa)$ of a family $\{X_\alpha:\alpha<\kappa\}$. We use the Frol\'ik sum to produce countably compact spaces with additional properties (like first countability) whose hyperspaces are not countably compact. We also prove that any product $\Pi_{\alpha<\kappa}H^0(X_\alpha)$ embeds into $H(F(X_\alpha:\alpha<\kappa))$.