The moduli problem is to describe the structure of the space
of isomorphism classes of Riemann surfaces of a given
topological type. This space is known as the moduli
space and has been at the center of pure mathematics for
more than a hundred years. In spite of its age, this field
still attracts a lot of attention, the smooth compact Riemann
surfaces being simply complex projective algebraic curves.
Therefore the moduli space of compact Riemann surfaces is also
the moduli space of complex algebraic curves. This space lies
on the intersection of many fields of mathematics and may be
studied from many different points of view.

The aim of this
monograph is to present information about the structure of the
moduli space using as concrete and elementary methods as
possible. This simple approach leads to a rich theory and
opens a new way of treating the moduli problem, putting new
life into classical methods that were used in the study of
moduli problems in the 1920s.