(MATH) We present two new algorithms for the problem of finding a minimum-cost k-vertex connected spanning subgraph. The first algorithm works on undirected graphs with at least 6k² vertices and achieves an approximation factor of 6 times the kth harmonic number, which is . The second algorithm works on directed and undirected graphs. It gives an $O(\sqrt{ n /\keps})$-approximation algorithm for any $\keps > 0$ and $k \le (1-\keps)n$. The latter algorithm also extends to other problems in network design with vertex connectivity requirements. Our main tools are setpair relaxations, a theorem of Mader's (in the undirected case) and iterative rounding (general case).