Perturbative superstring theory revisited

Perturbative superstring theory is revisited, with the goal of giving a simpler and more direct demonstration that multi-loop amplitudes are gauge-invariant (apart from known anomalies), satisfy space-time supersymmetry when expected, and have the ex- pected infrared behavior. The main technical tool is to make the whole analysis, including especially those arguments that involve integration by parts, on supermoduli space, rather than after descending to ordinary moduli space.