Based on Newton's law of Gravitation, there could be a negative (repulsive) gravitation force if the mass of the one the objects is negative, which is not possible to observe in our everyday life. However, recent observations of the expansion of our Universe have shown that it is expanding at an accelerating rate, a discovery that won a Nobel Prize in 2011. This accelerating expansion has been attributed to "dark energy" which is a currently poorly understood concept but has been found to make up the majority of our universe. This dark energy has a sort of repulsive or negative gravity effect which causes the expansion rate to accelerate. So it appears that there may be negative gravity, but it is still currently poorly understood.

Sort of. The rate of the expansion of the universe is increasing, which suggests that there is some kind of anti-gravity-like force. However, this spans the entire universe, and is equal across all space. We know of no local negative gravity fields.

I googled negative gravity and got this anti-gravity Wikipedia article:

"Anti-gravity is a theory of creating a place or object that is free from the force of gravity. It does not refer to the lack of weight under gravity experienced in free fall or orbit, or to balancing the force of gravity with some other force, such as electromagnetism or aerodynamic lift".