A signed graph is a pair (G,σ), where G=(V(G),E(G)) is an undirected graph and σ:E(G)→{+,−} is a function which marks each edge with “+” or “−”. Two signed graphs are equivalent if one of them can be changed to the other by a sequence of resigning operations. The single resigning operation chooses a vertex v∈V(G) and flips the signs of all edges incident to v. By [G,σ] we shall denote the equivalence class of the signed graph (G,σ). Each element of [G,σ] is called a presentation of [G,σ]. In this paper we shall call both (G,σ) and [G,σ] signed graphs. The coloring of signed graphs is defined through homomorphism. The signed graph [G,σ] is colored by the signed graph (G2,σ2), if there exists a presentation (G,σ1) of [G,σ] and a vertex-mapping ϕ from G to G2 which preserves signs of the edges. In this paper we show that: (a) The signed chromatic number for the class G of all 2-dimensional grids lies between 5 and 6. (b) Every signed grid with at most seven rows can be colored with five colors. (c) Every signed grid with two rows can be colored with four colors.