The noncommutative projective geometry is a name of several directions in noncommutative algebraic geometry. The noncommutative projective schemes are supposed to be represented with noncommutative -graded algebras; as in commutative case, different graded algebras may give rise to the isomorphic schemes. By definition the category of quasicoherent sheaves over the of a noncommutative graded algebra is the full category of (say left) modules modulo the torsion subcategory of the category of modules of finite length.
Most often one limits to noncommutative algebras which are close to commutative in certain sense. One very usable axiomatics is due Artin and Zhang and has been widely used: