Noncommutative (projective) algebraic geometry  seeks to use the techniques and intuition from  commutative algebraic geometry to study noncommutative   (graded) algebras and  related categories. The basic intuition here is that the noncommutative scheme (or, more formally, the category of coherent sheaves over this scheme) is nothing more than the category of graded modules over a noncommutative graded algebra modulo those of finite length. 

Under this intuition, irreducible noncommutative projective curves (equivalently graded domains with quadratic growth, like k[x,y]) have been classified, as have

NC projective planes  (suitable noncommutative analogues of k[x,y,z]).  A fundamental  open problem is to classify all noncommutative projective surfaces.

In this lecture I will  briefly surveying the general area and then describe some recent progress on the classification of noncommutative surfaces, notably through joint work with Dan Rogalski and  Sue  Sierra.