The k-rainbow domination problem is a well-known variation of the classical domination problem in graph theory. Motivated by applications in facility location problems, we study colorful k-rainbow domination, a variant of rainbow domination, where at least one color must be assigned to each vertex. Given a graph \(G=(V, E)\) and a set of k colors \(\{1, \dots , k\}\) we consider a function f that assigns a subset of colors to each vertex, that is \(f: V \rightarrow 2^{\{1, \dots , k\}} \) . If for each vertex v we have \(\bigcup _{x \in N[v]} f(x) = \{1, \dots , k\}\) and \(f(v) \ne \emptyset \) , then f is a colorful k-rainbow dominating function of G. The weight of f is \(w(f) = \sum _{v \in V}\vert f(v)\vert \) . The goal of the colorful k-rainbow domination problem is to find a colorful k-rainbow dominating function with the smallest weight. Such weight is called a colorful k-rainbow domination number. For \(k=2\) , we give a linear time algorithm that computes an optimal function. For \(k\ge 3\) , we show that the corresponding decision problem is NP-complete. Moreover, for \(k=3\) , it remains NP-complete if the input graph is restricted to be split or bipartite. Furthermore, we compute the colorful 3-rainbow domination number for cliques, bicliques, cycles, and paths. For trees and interval graphs, we give linear time algorithms that compute an optimal function.