In a simple digraph, a star of degree t is a union of t edges with a common tail. The k-domination number γ k (G) of digraph G is the minimum number of stars of degree at most k needed to cover the vertex set. We prove that γ k (T)= n/(k+1) when T is a tournament with n>=14klgk vertices. This improves a result of Chen, Lu and West. We also give a short direct proof of the result of E. Szekeres and G. Szekeres that every n-vertex tournament is dominated by at most lgn-lglgn+2 vertices.