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We review the current status of Finsler-Lagrange geometry andgeneralizations. The goal is to aid non-experts on Finsler spaces, butphysicists and geometers skilled in general relativity and particle theories,to understand the crucial importance of such geometric methods for applicationsin modern physics. We also would like to orient mathematicians working ingeneralized Finsler and Kahler geometry and geometric mechanics how they couldperform their results in order to be accepted by the community of ''orthodox''physicists.  Although the bulk of former models of Finsler-Lagrange spaces whereelaborated on tangent bundles, the surprising result advocated in our works isthat such locally anisotropic structures can be modelled equivalently onRiemann-Cartan spaces, even as exact solutions in Einstein and/or stringgravity, if nonholonomic distributions and moving frames of references areintroduced into consideration.  We also propose a canonical scheme when geometrical objects on a (pseudo)Riemannian space are nonholonomically deformed into generalized Lagrange, orFinsler, configurations on the same manifold. Such canonical transforms aredefined by the coefficients of a prime metric and generate target spaces asLagrange structures, their models of almost Hermitian/ Kahler, or nonholonomicRiemann spaces.  Finally, we consider some classes of exact solutions in string and Einsteingravity modelling Lagrange-Finsler structures with solitonic pp-waves andspeculate on their physical meaning.

Finsler and Lagrange Geometries in Einstein and String Gravity