Abstract: The Ricci flow of a manifold M is a flow on the manifold formed by the set of the riemannian metrics on M. This flow has become a great interest in Mathematics these last decades, in particular because of its leading role in Hamilton’s program to prove Thurston’s conjecture. When M is a unimodular three-dimensional Lie group, the restriction of its Ricci flow to the left-invariant metrics on M can be expressed in a very simple way and hence its behaviour becomes easy to study.