  Given a graph $G$, a geodesic packing in $G$ is a set of vertex-disjoint
maximal geodesics, and the geodesic packing number of $G$, ${\gpack}(G)$, is
the maximum cardinality of a geodesic packing in $G$. It is proved that the
decision version of the geodesic packing number is NP-complete. We also
consider the geodesic transversal number, ${\gt}(G)$, which is the minimum
cardinality of a set of vertices that hit all maximal geodesics in $G$. While
$\gt(G)\ge \gpack(G)$ in every graph $G$, the quotient ${\rm gt}(G)/{\rm
gpack}(G)$ is investigated. By using the rook's graph, it is proved that there
does not exist a constant $C < 3$ such that $\frac{{\rm gt}(G)}{{\rm
gpack}(G)}\le C$ would hold for all graphs $G$. If $T$ is a tree, then it is
proved that ${\rm gpack}(T) = {\rm gt}(T)$, and a linear algorithm for
determining ${\rm gpack}(T)$ is derived. The geodesic packing number is also
determined for the strong product of paths.
