  We give an affirmative answer to a long-standing conjecture of Thomassen,
stating that every sufficiently highly connected graph has a
$k$-vertex-connected orientation. We prove that a connectivity of order
$O(k^2)$ suffices. As a key tool, we show that for every pair of positive
integers $d$ and $t$, every $(t \cdot h(d))$-connected graph contains $t$
edge-disjoint $d$-rigid (in particular, $d$-connected) spanning subgraphs,
where $h(d) = 10d(d+1)$. This also implies a positive answer to the conjecture
of Kriesell that every sufficiently highly connected graph $G$ contains a
spanning tree $T$ such that $G-E(T)$ is $k$-connected.
