The main purpose of the paper is to provide an elementary proof of the following theorem: An infinite, non-degenerate, locally finite, homogeneous geometry is a projective or affine geometry over a finite field. This result was previously proved by \textit{G. Cherlin}, \textit{L. Harrington} and \textit{A. H. Lachlan} in Appendix 2 to their paper in Ann. Pure Appl. Logic 28, 103-135 (1985; Zbl 0566.03022) using the classification of finite primitive Jordan groups. Another proof of the theorem independent on the theory of Jordan groups can be found in the series of papers by \textit{B. I. Zil'ber} [see e.g. Sib. Math. J. 21, 219- 230 (1980); translation from Sib. Mat. Zh. 21, No.2, 98-112 (1980; Zbl 0486.03017); and Sov. Math., Dokl. 24, 149-151 (1981); translation from Dokl. Akad Nauk SSSR 259, 1039-1041 (1981; Zbl 0485.51004)]. The author's proof is elementary in the sense that the only point at which a substantial group-theoretic result is used is at the end of the proof, where the results of \textit{P. J. Cameron} and \textit{W. M. Kantor} [J. Algebra 60, 384-422 (1979; Zbl 0417.20044)] are used to show that a nondegenerate finite homogeneous geometry of sufficiently large dimensions is a projective or affine geometry over a finite field, possibly truncated. For further details the reader is referred to the paper itself.