  A general position set of a graph $G$ is a set of vertices $S$ in $G$ such
that no three vertices from $S$ lie on a common shortest path. In this paper we
introduce and study the general position achievement game. The game is played
on a graph $G$ by players A and B who alternatively pick vertices of $G$. A
selection of a vertex is legal if has not been selected before and the set of
vertices selected so far forms a general position set of $G$. The player who
selects the last vertex wins the game. Playable vertices at each step of the
game are described, and sufficient conditions for each of the players to win is
given. The game is studied on Cartesian and lexicographic products. Among other
results it is proved that A wins the game on $K_n\,\square\, K_m$ if and only
if both $n$ and $m$ are odd, and that B wins the game on $G\circ K_n$ if and
only if either B wins on $G$ or $n$ is even.
