[en] We show that the heterotic string is free of gauge and gravitational anomalies by showing that (a) unless the gauge group is E₈ x E₈ or Spin(32)/Z₂ or a subgroup, the internal sector partition function vanishes so there is no consistent theory; and (b) for E₈ x E₈ and Spin(32)/Z₂ compactifications, the longitudinal modes of the massless gauge particles decouple, as required by gauge invariance. We discuss the geometric interpretation for string theory when the action is invariant under a modular subgroup, and we develop the tools needed to study string world sheets with odd spin structures. (orig.)