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In this article, we review the current status of Finsler-Lagrange geometryand generalizations. 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 ''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 whengeometrical objects on a (pseudo) Riemannian space are nonholonomicallydeformed into generalized Lagrange, or Finsler, configurations on the samemanifold or on a corresponding tangent bundle. Such canonical transforms aredefined by the coefficients of a prime metric (it can be a solution of theEinstein equations) and generate target spaces as generalized Lagrangestructures, their models of almost Hermitian/ Kahler, or nonholonomic Riemannspaces with constant curvature, for some Finsler like connections. There areformulated the criteria when such constructions can be redefined equivalentlyin terms of the Levi Civita connection.

Finsler-Lagrange Geometries and Standard Theories in Physics: New Methods in Einstein and String Gravity