Let G = (V, E) be a graph and let k be a positive integer. A Roman k-dominating function ( R k-DF) on G is a function f : V(G) → {0, 1, 2} such that every vertex u for which f(u) = 0 is adjacent to at least k vertices v1, v2, …, vk with f(vi) = 2 for i = 1, 2, …, k. The weight of an R k-DF is the value f(V(G)) = ∑u∈V(G) f(u) and the minimum weight of an R k-DF on G is called the Roman k-domination number γkR(G) of G. In this paper, we present relations between γkR(G) and γR(G). Moreover, we give characterizations of some classes of graphs attaining equality in these relations. Finally, we establish a relation between γkR(G) and γR(G) for {K1,3, K1,3+e}-free graphs and we characterize all such graphs G with γkR(G) = γR(G)+t, where [Formula: see text].