the ricci iteration is a discrete analogue of the ricci flow. introduced in 2007, it has been studied extensively on kähler manifolds, providing a new approach to uniformisation. in the talk, we will define the ricci iteration on compact homogeneous spaces and discuss a number of existence, convergence and relative compactness results. this is largely based on joint work with timothy buttsworth (queensland), yanir rubinstein (maryland) and wolfgang ziller (penn).