  In these lectures, recent progress on multiloop superstring perturbation
theory is reviewed. A construction from first principles is given for an
unambiguous and slice-independent two-loop superstring measure on moduli space
for even spin structure. A consistent choice of moduli, invariant under local
worldsheet supersymmetry is made in terms of the super-period matrix. A variety
of subtle new contributions arising from a careful gauge fixing procedure are
taken into account.
  The superstring measure is computed explicitly in terms of genus two
theta-functions and reveals the importance of a new modular object of weight 6.
For given even spin structure, the measure exhibits a behavior under
degenerations of the worldsheet that is consistent with physical principles.
The measure allows for a unique modular covariant GSO projection. Under this
GSO projection, the cosmological constant, the 1-, 2- and 3- point functions of
massless supergravitons vanish pointwise on moduli space. A certain
disconnected part of the 4-point function is shown to be given by a convergent
integral on moduli space. A general consistent formula is given for the
two-loop cosmological constant in compactifications with central charge c=15
and with N=1 worldsheet supersymmetry. Finally, some comments are made on
possible extensions of this work to higher loop order.
