This text will acquaint the reader with the most recent advances in Finslerian geometries, i.e. anisotropic geometries, and their applications by the Japanese, European and American schools. It contains three introductory articles, one from each of these schools, giving a broad overview of basic ideas. Further papers treat topics from pure mathematics such as complex differential geometry, equivalence methods, Finslerian deformations, constant sprays, homogeneous contact transformations, Douglas spaces, submanifold theory, inverse problems, area theory, and more. This book completes the Kluwer trilogy on Finslerian Geometry by P.L. Antonelli and his associates. Audience: This volume will be of interest to physicists and mathematicians whose work involves quantum field theory, combination theory and relativity, programming and optimization. Mathematical biologists working in ecology and evolution will also find it useful.