  The period is a classical complex analytic invariant for a compact Riemann
surface defined by integration of differential 1-forms. It has a strong
relationship with the complex structure of the surface. In this chapter, we
review another complex analytic invariant called the harmonic volume. It is a
natural extension of the period defined using Chen's iterated integrals and
captures more detailed information of the complex structure. It is also one of
a few explicitly computable examples of complex analytic invariants. As an
application, we give an algorithm in proving nontriviality for a class of
homologically trivial algebraic cycles obtained from special compact Riemann
surfaces. The moduli space of compact Riemann surfaces is the space of all
biholomorphism classes of compact Riemann surfaces. The harmonic volume can be
regarded as an analytic section of a local system on the moduli space. It
enables a quantitative study of the local structure of the moduli space. We
explain basic concepts related to the harmonic volume and its applications of
the moduli space.
