The speed of gravitational waves is finite and very likely the speed of light. If a star (for instance the Sun) suddenly disappeared, we would not immediately feel anything. Assuming the speed of gravitational waves is the speed of light, during more or less eight minutes (in the case of the Sun), we would see the Sun and feel its gravitational field.
As long as Special Relativity and causality be exact concepts, nothing able to carry information can travel faster than light.
However, shadows and other things can travel faster than light. You can compute the speed of the spot of a laser that has been aimed at the surface of the Moon and you will realize that can be faster than light, but this spot cannot carry information.
Let me add that thanks to this effect (the finiteness of signal speed propagation), we can study the remote past of the universe (close to its origin) by means of looking at very distant objects.