In this paper, a new method to automatically discriminate the left and right eyes is proposed and validated utilizing a deep convolutional neural network. All parameters of the designed network are automatically estimated based on the characteristic differences between the left and right eye images. On the basis of the Alexnet network, the convolutional neural network designed in this paper consists of four convolutional layers and two fully connected layers, followed by a classifier serving as its last layer. According to our experimental results on a total of 42541 fundus images, the training accuracy of our network is about 100%, and the testing accuracy is as high as 99%. In addition, the proposed network is highly robust given that it successfully works for a large amount of fundus images with great variability. As far as we know, this is the first deep learning based network for left-vs-right eye discrimination that exhibits very high accuracy and precision.