  In this paper we introduce two new methods for constructing harmonic
morphisms from solvable Lie groups. The first method yields global solutions
from any simply connected nilpotent Lie group and from any Riemannian symmetric
space of non-compact type and rank $r\ge 3$. The second method provides us with
global solutions from any Damek-Ricci space and many non-compact Riemannian
symmetric spaces. We then give a continuous family of 3-dimensional solvable
Lie groups not admitting any complex valued harmonic morphisms, not even
locally.
