The Fourier domain is used in computer vision and machine learning as image analysis tasks in the Fourier domain are analogous to spatial domain methods but are achieved using different operations. Convolutional Neural Networks (CNNs) use machine learning to achieve state-of-the-art results with respect to many computer vision tasks. One of the main limiting aspects of CNNs is the computational cost of updating a large number of convolution parameters. Further, in the spatial domain, larger images take exponentially longer than smaller image to train on CNNs due to the operations involved in convolution methods. Consequently, CNNs are often not a viable solution for large image computer vision tasks. In this paper a Fourier Convolution Neural Network (FCNN) is proposed whereby training is conducted entirely within the Fourier domain. The advantage offered is that there is a significant speed up in training time without loss of effectiveness. Using the proposed approach larger images can therefore be processed within viable computation time. The FCNN is fully described and evaluated. The evaluation was conducted using the benchmark Cifar10 and MNIST datasets, and a bespoke fundus retina image dataset. The results demonstrate that convolution in the Fourier domain gives a significant speed up without adversely affecting accuracy. For simplicity the proposed FCNN concept is presented in the context of a basic CNN architecture, however, the FCNN concept has the potential to improve the speed of any neural network system involving convolution.