We compute the Einstein, Lorentz and Weyl worldsheet anomalies and the super-Poincare space-time anomalies of the heterotic superstring, using path-integrals and the gauge GAMMA-+ THETA = 0 to fix the local fermionic kappa-symmetry. The regulators for the various jacobians are all obtained from a recent general method of constructing consistent regulators. We do not use a special gauge for the metric. A general lemma for anomalies of algebraic and non-algebraic symmetries is obtained. Our results show that only for a certain class of metrics do all worldsheet and space-time anomalies cancel simultaneously after adding a suitable but complicated set of counterterms.