The book "Differential Geometry of Spray and Finsler Spaces" offers a comprehensive examination of sprays and Finsler metrics. It covers the fundamentals and the mathematical concepts associated with these topics, explaining how sprays are defined on manifolds as systems of compatible second-order ordinary differential equations. In contrast, Finsler metrics represent a family of norms in the tangent spaces that change smoothly with the base point. The work demonstrates that each Finsler metric defines a distance function through the length of minimal curves, allowing Finsler spaces to be regarded as regular metric spaces. The historical development of these concepts is also discussed, including the contributions of B. Riemann and P. Finsler to differential geometry. This academic book is aimed at students and professionals seeking a deeper understanding of differential geometry and its applications in modern mathematics.