Harmonic morphisms are maps $\phi:(M,g)\to (N,h)$ between Riemannian manifolds pulling back harmonic functions on N to harmonic functions on M. They have been characterised as being harmonic maps which are horizontally conformal. Under certain natural conditions they induce a conformal foliation on M with minimal leaves. In this talk we will give a short introduction to the general theory. Then we will described what is known in the important cases when M is a Lie group foliated in this way by the left translations of a semisimple subgroup of M.