A hypercomplex manifold is a triple \((M, J_1, J_2)\) consisting of a \(4n\)-dimensional manifold together with two anticommuting complex structures. This paper concerns the construction of non-compact homogeneous manifolds carrying such a structure; the compact case has been considered by \textit{D. D. Joyce} [J. Differ. Geom. 35, 743-761 (1992; Zbl 0782.53062)]. The authors construct such structures on a class of three step solvable groups. First, they consider the solvable Lie groups \(S\) corresponding to the rank one symmetric spaces of non-compact type and prove that all do admit \(S\)-invariant hypercomplex structures which are compatible with the symmetric metric, except when \(S\) corresponds to the complex hyperbolic space \(\mathbb{C} H^{2n}\). As a second topic, they consider in detail the Lie groups \(G\), with \(\dim{\mathcal G}'\leq 2\), \({\mathcal G}'=[{\mathcal G},{\mathcal G}]\), admitting hypercomplex structures; a classification of them when \(\dim{\mathcal G}'=1\) is given. If dim \({\mathcal G}'=2\) it is proved that, if \(G\) admits an invariant hypercomplex structure, then either \(G\) is two step nilpotent or \(G\) is locally isomorphic to \(\mathbb{R}^s\times (\mathbb{C}\rtimes\mathbb{C}^*)\).