Throughout the course of delivering a radiation therapy treatment, which may take several weeks, a patient's anatomy may change drastically, and adaptive radiation therapy (ART) may be needed. Cone-beam computed tomography (CBCT), which is often available during the treatment process, can be used for both patient positioning and ART re-planning. However, due to the prominent amount of noise, artifacts, and inaccurate Hounsfield unit (HU) values, the dose calculation based on CBCT images could be inaccurate for treatment planning. One way to solve this problem is to convert CBCT images to more accurate synthesized CT (sCT) images. In this work, we have developed a cycle-consistent generative adversarial network framework (CycleGAN) to synthesize CT images from CBCT images. This model is capable of image-to-image translation using unpaired CT and CBCT images in an unsupervised learning setting. The sCT images generated from CBCT through this CycleGAN model are visually and quantitatively similar to real CT images with decreased mean absolute error (MAE) from 69.29 HU to 29.85 HU for head-and-neck (H&N) cancer patients. The dose distributions calculated on the sCT by CycleGAN demonstrated a higher accuracy than those on CBCT in a 3D gamma index analysis with increased gamma index pass rate from 86.92% to 96.26% under 1 mm/1% criteria, when using the deformed planning CT image (dpCT) as the reference. We also compared the CycleGAN model with other unsupervised learning methods, including deep convolutional generative adversarial networks (DCGAN) and progressive growing of GANs (PGGAN), and demonstrated that CycleGAN outperformed the other two models. A phantom study has been conducted to compare sCT with dpCT, and the increase of structural similarity index from 0.91 to 0.93 shows that CycleGAN performed better than DIR in terms of preserving anatomical accuracy.