We propose a tensor-vector-scalar field theory of gravity by defining a corresponding Lagrange function. Deriving the field equations we observe, that the coupling to the tensor field gives mass to the vector field. After considerations of stability and convergence we develop the systems of differential equations for two cases. In the static case of a central mass we start with the Schwarzschild metric as a first approximation and get a behavior corresponding to the observation of galaxy rotation for sufficiently large distances from the center. Using this result a second approximation gives a qualitatively stronger gravity for very large distances, which could possibly describe the cohesion of galaxy clusters. When the distance tends to infinity, the gravity returns to a Newton-like decay due to the mass term of the vector field. In cosmology we find an expansion of a homogeneous, isotropic, flat universe at a lower rate than expected, but with a time contraction towards the past. Together these effects cause an observed redshift of light with far distant sources, which suggests an accelerated expansion of the universe.