  Following the new gauging fixing method of D'Hoker and Phong, we study
two-loop superstrings in hyperelliptic language. By using hyperelliptic
representation of genus 2 Riemann surface we derive a set of identities
involving the Szeg\"o kernel. These identities are used to prove the vanishing
of the cosmological constant and the non-renormalization theorem point-wise in
moduli space by doing the summation over all the 10 even spin structures.
Modular invariance is maintained at every stage of the computation explicitly.
The 4-particle amplitude is also computed and an explicit expression for the
chiral integrand is obtained. We use this result to show that the perturbative
correction to the $R^4$ term in type II superstring theories is vanishing at
two loops.
  In this paper, a summary of the main results is presented with detailed
derivations to be provided in two subsequent publications.
