  In the past 15 years a study of ``noncommutative projective geometry'' has
flourished. By using and generalizing techniques of commutative projective
geometry, one can study certain noncommutative graded rings and obtain results
for which no purely algebraic proof is known. For instance, noncommutative
graded domains of quadratic growth, or ``noncommutative curves,'' have now been
classified by geometric data and these rings must be noetherian. Rings of cubic
growth, or ``noncommutative surfaces,'' are not yet classified, but a rich
theory is currently forming. In this survey, we describe some of these results
and examine the question of which rings should be included in noncommutative
projective geometry.
