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Abstract: We present the gravitational dynamics with generalized field equations in Finsler and Finsler-like geometries. In this approach, the gravitational field is considered in the framework of extra dimensions with more degrees of freedom. The Einstein‐Finsler‐like gravity theories can be considered as natural candidates for the investigation of Lorentz violation, local anisotropies and provide a geometrical background for the emergence of dark matter and dark energy. In addition, modified gravity theories are extended in the framework of a geometry of an enriched spacetime with extra degrees of freedom. Friedmann equations are derived in a locally-anisotropic form of spacetime. In such theories, the concept of nonlinear connection, which connect the external and internal structures of spacetime, plays a crucial role. Some cosmological consequences are shown for the Finsler-Randers, Schwarzschild-Finsler-Randers models and generalized scalar-tensor theories.