Frequent occurrence of electrooculography (EOG) artifacts leads to serious problems in interpreting and analyzing the electroencephalogram (EEG). In this paper, a robust method is presented to automatically eliminate eye-movement and eye-blink artifacts from EEG signals. Independent Component Analysis (ICA) is used to decompose EEG signals into independent components. Moreover, the features of topographies and power spectral densities of those components are extracted to identify eye-movement artifact components, and a support vector machine (SVM) classifier is adopted because it has higher performance than several other classifiers. The classification results show that feature-extraction methods are unsuitable for identifying eye-blink artifact components, and then a novel peak detection algorithm of independent component (PDAIC) is proposed to identify eye-blink artifact components. Finally, the artifact removal method proposed here is evaluated by the comparisons of EEG data before and after artifact removal. The results indicate that the method proposed could remove EOG artifacts effectively from EEG signals with little distortion of the underlying brain signals.