Just like many other topics in computer vision, image classification has achieved significant progress recently by using deep-learning neural networks, especially the Convolutional Neural Networks (CNN). Most of the existing works are focused on classifying very clear natural images, evidenced by the widely used image databases such as Caltech-256, PASCAL VOCs and ImageNet. However, in many real applications, the acquired images may contain certain degradations that lead to various kinds of blurring, noise, and distortions. One important and interesting problem is the effect of such degradations to the performance of CNN-based image classification. More specifically, we wonder whether image-classification performance drops with each kind of degradation, whether this drop can be avoided by including degraded images into training, and whether existing computer vision algorithms that attempt to remove such degradations can help improve the image-classification performance. In this paper, we empirically study this problem for four kinds of degraded images -hazy images, underwater images, motion-blurred images and fish-eye images. For this study, we synthesize a large number of such degraded images by applying respective physical models to the clear natural images and collect a new hazy image dataset from the Internet. We expect this work can draw more interests from the community to study the classification of degraded images.