This work presents TorchRadon -- an open source CUDA library which contains a set of differentiable routines for solving computed tomography (CT) reconstruction problems. The library is designed to help researchers working on CT problems to combine deep learning and model-based approaches. The package is developed as a PyTorch extension and can be seamlessly integrated into existing deep learning training code. Compared to the existing Astra Toolbox, TorchRadon is up to 125 faster. The operators implemented by TorchRadon allow the computation of gradients using PyTorch backward(), and can therefore be easily inserted inside existing neural networks architectures. Because of its speed and GPU support, TorchRadon can also be effectively used as a fast backend for the implementation of iterative algorithms. This paper presents the main functionalities of the library, compares results with existing libraries and provides examples of usage.