We propose a novel approach to translate unpaired contrast computed tomography (CT) scans to non-contrast CT scans and the other way around. Solving this task has two important applications: (i) to automatically generate contrast CT scans for patients for whom injecting contrast substance is not an option, and (ii) to enhance alignment between contrast and non-contrast CT by reducing the differences induced by the contrast substance before registration. Our approach is based on cycle-consistent generative adversarial convolutional transformers, for short, CyTran. Our neural model can be trained on unpaired images, due to the integration of a cycle-consistency loss. To deal with high-resolution images, we design a hybrid architecture based on convolutional and multi-head attention layers. In addition, we introduce a novel data set, Coltea-Lung-CT-100W, containing 3D triphasic lung CT scans (with a total of 37,290 images) collected from 100 female patients. Each scan contains three phases (non-contrast, early portal venous, and late arterial), allowing us to perform experiments to compare our novel approach with state-of-the-art methods for image style transfer. Our empirical results show that CyTran outperforms all competing methods. Moreover, we show that CyTran can be employed as a preliminary step to improve a state-of-the-art medical image alignment method. We release our novel model and data set as open source at: https://github.com/ristea/cycle-transformer. Copyright © 2021, The Authors. All rights reserved.