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) $ ightarrow$ {0, 1, 2} such that every vertex u for which f(u) = 0 is adjacent to at least k vertices $upsilon_1,;upsilon_2,;{ldots},;upsilon_k$ with $f(upsilon_i)$ = 2 for i = 1, 2, $ldot$, k. The weight of a Roman k-dominating function is the value f(V (G)) = $sum_{u{in}v(G)}$ f(u). The minimum weight of a Roman k-dominating function on a graph G is called the Roman k-domination number ${gamma}_{kR}$(G) of G. Note that the Roman 1-domination number $gamma_{1R}$(G) is the usual Roman domination number $gamma_R$(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.