Medical image registration plays a crucial role in diagnosis, treatment planning, and anatomical studies. Classical methods, relying on iterative optimization algorithms, are complex and computationally intensive. Recent advances in deep learning, particularly with convolutional neural networks (CNNs) like VoxelMorph, have shown promise. However, they often yield non-smooth deformation fields and require inference at the same image resolution as the training data. To overcome these challenges, we introduce FNOReg, a novel model based on Fourier Neural Operators (FNOs), which can be trained on reduced-resolution images without quality loss and produces smoother deformation fields. We evaluated FNOReg on 2D and 3D datasets, demonstrating comparable quality to popular models like VoxelMorph, Fourier-Net, and TransMorph when trained at the same resolution. However, these models exhibit significant quality decreases of up to 24.9% for 2D and 24.6% for 3D data, when trained at halved resolutions. In contrast, FNOReg demonstrates only marginal quality decreases of up to 0.8% for 2D and 2.7% for 3D data. This flexibility is essential for efficiently handling large image resolutions, particularly in 3D imaging. Moreover, FNOReg produces smoother deformation fields. The code is available at https://github.com/anac0der/fnoreg .