  A covariant scalar-tensor-vector gravity theory is developed which allows the
gravitational constant $G$, a vector field coupling $\omega$ and the vector
field mass $\mu$ to vary with space and time. The equations of motion for a
test particle lead to a modified gravitational acceleration law that can fit
galaxy rotation curves and cluster data without non-baryonic dark matter. The
theory is consistent with solar system observational tests. The linear
evolutions of the metric, vector field and scalar field perturbations and their
consequences for the observations of the cosmic microwave background are
investigated.
