  In this contribution to the Proceedings of the Conference on Analysis,
Complex Geometry, and Mathematical Physics, an expository overview of
superstring perturbation theory to two loop order is presented to an audience
of mathematicians and physicists. Recent results on perturbative supersymmetry
breaking effects in Heterotic string theory compactified on Z_2 \times Z_2
Calabi-Yau orbifolds, and the calculation of the two-loop vacuum energy in
these theories are discussed in detail, and the appearance of a new modular
identity with respect to Sp(4,Z)/Z_4 is reviewed.
