  The general position number ${\rm gp}(G)$ of a connected graph $G$ is the
cardinality of a largest set $S$ of vertices such that no three distinct
vertices from $S$ lie on a common geodesic; such sets are refereed to as
gp-sets of $G$. The general position number of cylinders $P_r\,\square\, C_s$
is deduced. It is proved that ${\rm gp}(C_r\,\square\, C_s)\in \{6,7\}$
whenever $r\ge s \ge 3$, $s\ne 4$, and $r\ge 6$. A probabilistic lower bound on
the general position number of Cartesian graph powers is achieved. Along the
way a formula for the number of gp-sets in $P_r\,\square\, P_s$, where $r,s\ge
2$, is also determined.
