We shall describe an ODE for a curve of Lie brackets whose solutions are, up to pullback by time-dependent diffeomorphisms, the Ricci flows starting at any homogeneous Riemannian manifold. As an application, one obtains that the Ricci flow starting at any metric on the euclidean space which is invariant by a transitive solvable Lie group (i.e. solvmanifolds) is immortal. If the group is nilpotent (i.e. nilmanifolds), then in addition the curvature is O(1/t)), and up to pull-back by time-dependent invertible linear maps, the Ricci flow is the negative gradient flow of a degree 4 homogeneous polynomial (the square norm of the Ricci tensor), converges in C^\infty to the flat metric uniformly on compact sets, and up to rescaling (scalar curvature = -1), converges in C^\infty to a Ricci soliton metric uniformly on compact sets. Concerning Ricci solitons, we will define algebraic solitons by generalizing the concept of nilsoliton and give an overview of what is known in the homogeneous case.