  The field equations for Scalar-Tensor-Vector-Gravity (STVG) or modified
gravity (MOG) have a static, spherically symmetric black hole solution
determined by the mass $M$ with two horizons. The strength of the gravitational
constant is $G=G_N(1+\alpha)$ where $\alpha$ is a parameter. A regular
singularity-free MOG solution is derived using a nonlinear field dynamics for
the repulsive gravitational field component and a reasonable physical
energy-momentum tensor. The Kruskal-Szekeres completion of the MOG black hole
solution is obtained. The Kerr-MOG black hole solution is determined by the
mass $M$, the parameter $\alpha$ and the spin angular momentum $J=Ma$. The
equations of motion and the stability condition of a test particle orbiting the
MOG black hole are derived, and the radius of the black hole photosphere and
the shadows cast by the Schwarzschild-MOG and Kerr-MOG black holes are
calculated. A traversable wormhole solution is constructed with a throat
stabilized by the repulsive component of the gravitational field.
