Although no known particles have negative mass, physicists (primarily Robert L. Forward) have been able to describe some of the anticipated properties such particles may have. Negative mass would produce "negative gravity" that repels ordinary positive mass, but would be attracted to positive mass and other negative mass particles in a normal matter.
This behaviour is bizarre, in that it's completely inconsistent with our 'normal universe' common sense expected behaviour from working with positive masses, but it is completely mathematically consistent and introduces no apparent contradictions when physics analysis is performed on the behaviours.
Naïve first impressions are that this violates conservation of momentum and/or energy, but in fact if the masses are equal in magnitude, one being of positive value and the other negative, then the momentum of the system is zero if they both travel together and accelerate together, no matter what speed:
Forward also showed that if m(-) and m(+) are not the same magnitude of mass, the equations are still consistent.
Some of the behaviours this seems to introduce are really bizarre, such as a comingled positive matter gas and negative matter gas having the positive matter portion increase in temperature without bound. However, the negative matter portion gains negative temperature at the same rate, again balancing out.
Forward has proposed a design for spacecraft propulsion using negative mass that requires no energy input and no reaction mass to achieve arbitrarily high acceleration, though of course a major obstacle to the construction of such a spacecraft is the fact that negative mass remains purely hypothetical.
The term Exotic matter is also casually attached to any material which would be difficult to produce (such as metallic hydrogen or a Bose-Einstein condensate) or which exhibits unusual properties (such as fullerenes or nanotubes), even though these materials are relatively mundane in their composition. It could also refer to material composed of some form of exotic atom.