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A 2-rainbow dominating function (2RDF) g : V → P(A) ( where P(A) is the power set of the set of two colors A = {1, 2} ) of a graph G = (V,E) is defined to be satisfying the condition that for every vertex v ∊ V with g(v) = ϕ we have S u∊ℕ(v) g(u) = A. The minimum value of w(g) = Σ v∊V j∣(v)∣ among all such functions g of G is called the 2-rainbow domination number of G and is denoted by γr2 (G). A set S ⊆ V is a global dominating set of a graph G if S dominates both G and its complement G. The minimum cardinality γg(G) of a global dominating set of G is called the global domination number of the graph G. In this paper, we introduce the global 2-rainbow domination number γgr2 (G) of a graph G, study some of its properties, determine its exact values for some specific graphs and we characterize the graphs G with γgr2 (G) = p, where p is the number of vertices of G. [ABSTRACT FROM AUTHOR]