This paper addresses the problem of the classification of hyperspectral remote sensing images by support vector machines (SVMs). First, we propose a theoretical discussion and experimental analysis aimed at understanding and assessing the potentialities of SVM classifiers in hyperdimensional feature spaces. Then, we assess the effectiveness of SVMs with respect to conventional feature-reduction-based approaches and their performances in hypersubspaces of various dimensionalities. To sustain such an analysis, the performances of SVMs are compared with those of two other nonparametric classifiers (i.e., radial basis function neural networks and the K-nearest neighbor classifier). Finally, we study the potentially critical issue of applying binary SVMs to multiclass problems in hyperspectral data. In particular, four different multiclass strategies are analyzed and compared: the one-against-all, the one-against-one, and two hierarchical tree-based strategies. Different performance indicators have been used to support our experimental studies in a detailed and accurate way, i.e., the classification accuracy, the computational time, the stability to parameter setting, and the complexity of the multiclass architecture. The results obtained on a real Airborne Visible/Infrared Imaging Spectroradiometer hyperspectral dataset allow to conclude that, whatever the multiclass strategy adopted, SVMs are a valid and effective alternative to conventional pattern recognition approaches (feature-reduction procedures combined with a classification method) for the classification of hyperspectral remote sensing data.