This article may be too technical for most readers to understand.(June 2020)
In more speculative theories, negative energy is involved in time travel to the past, creation of artificial wormholes, which may also allow for time travel, Krasnikov tubes, Alcubierre drive, and potentially other types of warp drives for faster-than-light space travel.
The positive gravitational attraction between two massive objects is accompanied by a negative amount of gravitational potential in the field which attracts them. As the distance between them approaches infinity, the gravitational attraction approaches zero from the positive side of the real number line and the gravitational potential approaches zero from the negative side. Therefore, as two massive objects move towards each other, the motion accelerates under gravity causing an increase in the (positive) kinetic energy of the system and an increase of the same amount in the (negative) gravitational potential energy of the object. This is because the law of conservation of energy requires that the net energy of the system will not change. The gravitational potential energy is a kind of binding energy. 
A universe in which positive energy dominates will eventually collapse in a "Big Crunch", while an "open" universe in which negative energy dominates will either expand indefinitely or eventually disintegrate in a "big rip". In the zero-energy universe model ("flat" or "Euclidean"), the total amount of energy in the universe is exactly zero: its amount of positive energy in the form of matter is exactly cancelled out by its negative energy in the form of gravity. (It is unclear which, if any, of these models accurately describes the real universe.)
In quantum theory, the uncertainty principle allows the vacuum of space to be filled with virtual particle-antiparticle pairs which appear spontaneously and exist for only a short time before, typically, annihilating themselves again. Some of these virtual particles can have negative energy. Their behaviour plays a role in several important phenomena, as described below.
In the Casimir effect, two flat plates placed very close together restrict the wavelengths of quanta which can exist between them. This in turn restricts the types and hence number and density of virtual particle pairs which can form in the intervening vacuum and can result in a negative energy density. This causes an attractive force between the plates, which has been measured.
It is possible to arrange multiple beams of laser light such that destructive quantum interference suppresses the vacuum fluctuations. Such a squeezed vacuum state involves negative energy. The repetitive waveform of light leads to alternating regions of positive and negative energy.
According to the theory of the Dirac sea, developed by Paul Dirac in 1930, the vacuum of space is full of negative energy. This theory was developed to explain the anomaly of negative-energy quantum states predicted by the Dirac equation.
The Dirac sea theory correctly predicted the existence of antimatter two years prior to the discovery of the positron in 1932 by Carl Anderson. However, the Dirac sea theory treats antimatter as a hole where there is the absence of a particle rather than as a real particle. Quantum field theory (QFT), developed in the 1930s, deals with antimatter in a way that treats antimatter as made of real particles rather than the absence of particles, and treats a vacuum as being empty of particles rather than full of negative-energy particles like in the Dirac sea theory.
Quantum field theory has displaced the Dirac sea theory as a more popular explanation of these aspects of physics. Both the Dirac sea theory and quantum field theory are equivalent by means of a Bogoliubov transformation, so the Dirac sea can be viewed as an alternative formulation of quantum field theory, and is thus consistent with it.
The intense gravitational fields around black holes create phenomena which are attributed to both gravitational and quantum effects. In these situations, a particle's Killing vector may be rotated such that its energy becomes negative.
Virtual particles can exist for a short period. When a pair of such particles appears next to a black hole's event horizon, one of them may get drawn in. This rotates its Killing vector so that its energy becomes negative and the pair have no net energy. This allows them to become real and the positive particle escapes as Hawking radiation, while the negative-energy particle reduces the black hole's net energy. Thus, a black hole may slowly evaporate.
For a rotating black hole, the rotation creates an ergosphere outside the event horizon. Since the ergosphere is outside the event horizon, particles can escape from it. Within the ergosphere, a particle's Killing vector may be rotated to give it negative energy. The negative-energy particle then crosses the event horizon into the black hole, with the law of conservation of energy requiring that an equal amount of positive energy should escape. This is thought to be how the intense radiation emitted by quasars is generated.
Negative energy appears in the speculative theory of wormholes, where it is needed to keep the wormhole open. A wormhole directly connects two locations which may be separated arbitrarily far apart in both space and time, and in principle allows near-instantaneous travel between them. However physicists such as Roger Penrose regard such ideas as unrealistic, more fiction than speculation.
A theoretical principle for a faster-than-light (FTL) warp drive for spaceships has been suggested, involving negative energy. The Alcubierre drive comprises a solution to Einstein's equations of general relativity, in which a bubble of spacetime is moved rapidly by expanding space behind it and shrinking space in front of it.