  For a positive integer $k$ we consider the $k$-vertex-connectivity game,
played on the edge set of $K_n$, the complete graph on $n$ vertices. We first
study the Maker-Breaker version of this game and prove that, for any integer $k
\geq 2$ and sufficiently large $n$, Maker has a strategy for winning this game
within $\lfloor k n/2 \rfloor + 1$ moves, which is clearly best possible. This
answers a question of Hefetz, Krivelevich, Stojakovi\'c and Szab\'o. We then
consider the strong $k$-vertex-connectivity game. For every positive integer
$k$ and sufficiently large $n$, we describe an explicit first player's winning
strategy for this game.
