For two vertices u and v of a connected graph G, the set IG[u; v] consists of all those vertices lying on u v geodesics in G. Given a set S of vertices of G, the union of all sets IG[u; v] for u; v 2 S is denoted by IG[S]. A set S V (G) is a geodetic set if IG[S] = V (G) and the minimum cardinality of a geodetic set is its geodetic number g(G) of G. Bounds for the geodetic number of strong product graphs are obtainted and for several classes improved bounds and exact values are obtained.