Electron impact ionization of C-60 and C-70 was studied using a molecular/electron beam ion source in combination with a two sector held mass spectrometer operated in the ion beam deflection mode. Relative partial ionization cross sections for the production of singly and multiply charged parent ions (up to charge state z = 4) and fragment ions (down to C-44(+) in the case of C-60 and down to C-50(2+) in the case of C-70) were determined from threshold up to 1000 eV electron energy. Absolute partial and total ionization cross sections are obtained using a novel approach for the absolute calibration involving an intercomparison of the cation with the anion yield. The results obtained reveal not only an anomalous large parent ion cross section as compared to other ionization channels [e.g., sigma(C-60(+)/C-60) is more than a factor of 30 larger than sigma(C-58(+)/C-60)] but also anomalies for the production of multiply charged parent and fragment ions. For instance, the maximum cross section for the formation of C-60(2+) amounts to 30% of the maximum C-60(+) cross section and that of C-70(2+) to about 50% of C-70(+). Moreover, for all fragment ions, the formation of the doubly charged fragment ions has a larger cross section than that of the respective singly charged fragment ion. These peculiar features of the kinetics of electron impact ionization of C-60 and C-70 are related to the specific electronic and geometric structures of these fullerenes. The present absolute cross-section data for the summed up partial cross sections are in good agreement with a recent semiclassical calculation from our laboratory.