A vertex-coloring of a graph G is said to be locally identifying if for any pair (u,v) of adjacent vertices of G, with distinct closed neighborhood, the set of colors that appears in the closed neighborhoods of u and v are distinct. In this paper, we give several bounds on the minimum number of colors needed in such a coloring for different families of graphs (planar graphs, some subclasses of perfect graphs, graphs with bounded maximum degree) and prove that deciding whether a subcubic bipartite graph with large girth has a locally identifying coloring with 3 colors is an NP-complete problem.