From Wikipedia

The Case For Antigravity

In physical cosmology, astronomy and celestial mechanics, anti-gravity is the idea of creating a place or object that is free from the force of gravity. It does not refer to the lack of weight (but not the lack of gravity) experienced in free fall or orbit, nor to balancing the force of gravity with some other force, such as electromagnetism (or simply aerodynamic lift).

Instead, anti-gravity requires that the fundamental causes of the force of gravity be made either not present or not applicable to the place or object through some kind of technological intervention. Anti-gravity is a recurring concept in science fiction, particularly in the context of spacecraft propulsion. The concept was first introduced formally as "Cavorite" in H. G. Wells' The First Men in the Moon, and has been a favorite item of imaginary technology since that day.

In the first mathematically accurate description of gravity, Newton's law of universal gravitation, gravity was an external force transmitted by unknown means. However in the early part of the 20th century Newton's model was replaced by the more general and complete description known as general relativity. In general relativity, gravity is not a force in the traditional sense of the word, but the result of the geometry of space itself. These geometrical solutions always cause attractive "forces". Under general relativity, anti-gravity is highly unlikely, except under contrived circumstances that are regarded as unlikely or impossible. The term "anti-gravity" is also sometimes used to refer to hypothetical reactionless propulsion drives based on certain solutions to general relativity, although these do not oppose gravity as such.

There are numerous newer theories that add onto general relativity or replace it outright, and some of these appear to allow anti-gravity-like solutions. However, according to the current widely accepted physical theories, verified in experiments, and according to the major directions of physical research, it is considered highly unlikely that anti-gravity is possible.