  Complex geometry and supergeometry are closely entertwined in superstring
perturbation theory, since perturbative superstring amplitudes are formulated
in terms of supergeometry, and yet should reduce to integrals of holomorphic
forms on the moduli space of punctured Riemann surfaces. The presence of
supermoduli has been a major obstacle for a long time in carrying out this
program. Recently, this obstacle has been overcome at genus 2, which is the
first loop order where it appears in all amplitudes. An important ingredient is
a better understanding of the relation between geometry and supergeometry, and
between holomorphicity and superholomorphicity. This talk provides a survey of
these developments and a brief discussion of the directions for further
investigation.
