Let k be a positive integer, and let G be a simple graph with vertex set V (G). A Roman k-dominating function 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

with

= 2 for i = 1, 2,

, k. The weight of a Roman k-dominating function is the value f(V (G)) =

f(u). The minimum weight of a Roman k-dominating function on a graph G is called the Roman k-domination number

(G) of G. Note that the Roman 1-domination number

(G) is the usual Roman domination number

(G). In this paper, we investigate the properties of the Roman k-domination number. Some of our results extend these one given by Cockayne, Dreyer Jr., S. M. Hedetniemi, and S. T. Hedetniemi [2] in 2004 for the Roman domination number.