  We analyze the foundations of Finsler gravity theories with metric compatible
connections constructed on nonholonomic tangent bundles, or (pseudo) Riemannian
manifolds. There are considered "minimal" modifications of Einstein gravity
(including theories with violation of local Lorentz invariance) and shown how
the general relativity theory and generalizations can be equivalently
re--formulated in Finsler like variables. We focus on Finsler branes solutions
and perspectives in modern cosmology.
