The authors give a combinatorial characterization of the incidence structures whose points and blocks are the h- and \((h+1)\)-dimensional affine and projective spaces \(A_{h,h+1}({\mathbb{A}})\), \(P_{h,h+1}({\mathbb{P}})\), respectively. They define the concepts of affine and projective G-spaces, examples of which are the above mentioned incidence structures, by suitably and furtherly axiomatizing the class of semilinear spaces (P,L) [cf. \textit{A. P. Sprague}, Discrete Math. 33, 79- 87 (1981; Zbl 0461.51003)], and by determining several properties of this subclass. In particular, that an affine G-space does have a natural projective G-extension. Theorem I: A projective G-space of finite index h is isomorphic to a \(P_{r-h-1,r-h}({\mathbb{P}})\), for some projective space \({\mathbb{P}}.\) Theorem II: Let (P,L) be an affine G-space of finite index. Then, there exists an affine space A and an integer h such that (P,L) is isomorphic to \(A_{h,h+1}({\mathbb{A}})\).