The zeta-function method for regularizing determinants is used to calculate the chiral anomalies of several field-theory models. In SU(N) gauge theories without ${\ensuremath{\gamma}}_{5}$ couplings, the results of perturbation theory are obtained in an unambiguous manner for the full gauge theory as well as for the corresponding external-field problem. If axial-vector couplings are present, different anomalies occur for the two cases. The result for the full gauge theory is again uniquely determined; for its nongauge analog, however, ambiguities can arise. The connection between the basic path integral and the operator used to construct the heat kernel is investigated and the significance of its Hermiticity and gauge covariance are analyzed. The implications of the Wess-Zumino conditions are considered.