Function spaces of Hardy–Sobolev–Besov type on symmetric spaces of noncompact type and unimodular Lie groups are investigated. The spaces were originally defined by uniform localization. In the paper we give a characterization of the spaceFsp,q(X) andBsp,q(X) in terms of heat and Poisson semigroups, for 1⩽p,q⩽∞ ands∈R. The main tool we use, is an atomic decomposition of function spaces on manifolds.