Finsler geometry has been a main field of differential geometry in the 20th and 21st century. Its most natural generalizations are spray geometry and the geometry of generalized metric (Riemannian) spaces. This volume contains the authors own results embedded in the solid framework of classical results. Wherever possible, the author uses a modern, index- and coordinate-free apparatus. An Appendix, however, contains the formulations of several geometric objects presented in the text in terms of coordinates as well. The author quotes the most important original sources as well as the treatises...