Two vertices u,v of connected graph G are doubly resolved by x,y\in V(G)if d(v; x)-d(u; x)\neq d(v; y)-d(u; y): A set W of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of W. ψ(G) is the minimum cardinality of a doubly resolving set for the graph G. The aim of this paper is to investigate doubly resolving sets in graphs. An upper bound for Ψ(G) is obtained in terms of order and diameter of G. ψ(G) is computed for some important graphs and all graphs G of order n with the property ψ(G)=n-1 are characterized.