  In this survey article we describe some geometric results in the theory of
noncommutative rings and, more generally, in the theory of abelian categories.
  Roughly speaking and by analogy with the commutative situation, the category
of graded modules modulo torsion over a noncommutative graded ring of
quadratic, respectively cubic growth should be thought of as the noncommutative
analogue of a projective curve, respectively surface. This intuition has lead
to a remarkable number of nontrivial insights and results in noncommutative
algebra. Indeed, the problem of classifying noncommutative curves (and
noncommutative graded rings of quadratic growth) can be regarded as settled.
Despite the fact that no classification of noncommutative surfaces is in sight,
a rich body of nontrivial examples and techniques, including blowing up and
down, has been developed.
