Mathematically speaking, perturbative string theory is the study of certain natural functors on the category of Riemann surfaces or complex curves. This relationship is easily explained. A fundamental string is a loop in space. If we consider the history of such a string in space-time, it will sweep out a two-dimensional surface known as a world-sheet. The internal consistency of the theory demands that this world-sheet carries a complex structure and thus becomes a Riemann surface. For a free propagating string the topology of this worldsheet is a cylinder, but if we include interactions, under which the string can split and join, the surfaces can have arbitrary topology and represent the possible classical trajectories of a collection of interacting strings.
It is a very general principle that in any quantum theory we should
associate an amplitude to any possible trajectory. Concretely this means
that if we have a Riemann surface 
with, say, n incoming
boundaries and m outgoing boundaries, as depicted in figure 1, there
will be a linear map 
associated to the surface, where 
is the Hilbert
space describing the posssible quantum states of a single string. These
maps are not arbitrary, but satisfy all kinds of consistency relations,
making it into a functor from the category of Riemann surfaces to the
category of Hilbert spaces. Examples of such functors can be defined by
maps of the surface into a complex manifold, explaining the
relevance to the problem of counting curves in algebraic geometry. The
simple existence of the `pair of pants' surface, the three-holed sphere,
immediately tells us that there is some kind of natural algebraic
structure 
indicating
the relation with representation theory.
Since string theory is a quantum theory, Feynman's principle of sums
over histories tells us to consider not only one particular surface but
to sum over all possible topologies, i.e. all genera
, and integrate over all possible complex structures. In this
way we are led to consider formal expressions as
where we integrate the amplitudes 
over the moduli space 
of Riemann surfaces of genus g. Here the string coupling
constant or Planck's constant 
controls the perturbative quantisation; the classical
theory is recovered in the limit 
where only
spherical topologies contribute.
Unfortunately, straightforward estimates show that the above expression
for 
can be at most an asymptotic expansion. So the
definition of string theory in terms of Riemann surfaces is at least
incomplete. This is a well-known phenomenon in quantum field theories,
where the Feynman diagrams rarely capture the full story. To add insult
to injury, there furthermore seem to be five independent perturbative
string theories, i.e. consistent functors , that differ
rather dramatically in their basic properties, destroying the uniqueness
of the theory.