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.