Let k≥1 be an integer, and let G be a graph. A {\it
k-rainbow dominating function} (or a {\it k-RDF}) of G is a
function f from the vertex set V(G) to the family of all subsets
of {1,2,…,k} such that for every v∈V(G) with
f(v)=∅, the condition ⋃u∈NG(v)f(u)={1,2,…,k} is fulfilled, where NG(v) is
the open neighborhood of v. The {\it weight} of a k-RDF f of
G is the value ω(f)=∑v∈V(G)∣f(v)∣. A k-rainbow
dominating function f in a graph with no isolated vertex is called
a {\em total k-rainbow dominating function} if the subgraph of G
induced by the set {v∈V(G)∣f(v)=∅} has no isolated
vertices. The {\em total k-rainbow domination number} of G, denoted by
γtrk(G), is the minimum weight of a total k-rainbow
dominating function on G. The total 1-rainbow domination is the
same as the total domination. In this paper we initiate the
study of total k-rainbow domination number and we investigate its
basic properties. In particular, we present some sharp bounds on the
total k-rainbow domination number and we determine the total
k-rainbow domination number of some classes of graphs.