Let G be a graph with vertex set V. A function f : V→{−1 ,0 ,2} is called a Roman balanced dominating function (RBDF) of G if ∑ u ∈ N G [ v ] f ( u ) = 0 for each vertex v∈ V. The maximum (resp. minimum) Roman balanced number γ Rb M ( G ) (resp. γ Rb m ( G ) ) is the maximum (resp. minimum) value of ∑ v ∈ V f ( v ) among all Roman balanced dominating functions f. A graph G is called Rd-balanced if γ Rb M ( G ) = γ Rb m ( G ) = 0 . We present several lower and upper bounds on γ Rb M ( G ) and γ Rb m ( G ) and further determine several classes of Rd-balanced graphs.