An Introduction to Riemann-Finsler Geometry
by D. Bao, S.S Chern, and Z. Shen

In this book, the reader will find a comprehensive account of that part of Finsler geometry that is approachable by Riemannian and infinitesimal techniques. Among the topics covered, we find the Chern connection and the theorems of Gauss-Bonnet, Hopf-Rinow, Bonnet-Myers, and Cartan-Hadamard. There are many examples of symmetric and non-symmetric Finsler spaces with explicit formulas and calculations.

Differential Geometry of Spray and Finsler Geometry
by Z. Shen

In this work, Shen studies Finsler manifolds, manifolds provided with a second order ODE (i.e., a spray), as well as Finsler manifolds provided with a volume form as a supplementary geometric structure. The book is a good survey of infinitesimal techniques in Finsler and path geometry. It also contains some new results and introduces a new invariant, the S curvature, which together with the curvature controls the volume of geodesic balls on Finsler spaces.

Unfortunately, because of copyright constraints, these books can no longer be obtained electronically.

Geometric meanings of cuvatures in Finsler geometry
by Z. Shen

This paper is a well-written survey of the different notions of curvature in Finsler geometry and their relation to Minkowski spaces, behaviour of geodesics, and comparison geometry.

Volumes in normed and Finsler spaces
by J.C. Álvarez Paiva and A. Thompson

This paper is a survey of the theory of volumes and area integrands in normed and Finsler spaces. It contains a (hopefully) comprehensible presentation of Busemann's theory of convexity on Grassmannians that should be of interest to people working on geometric measure theory and variational calculus.