This paper presents an end-to-end learning-based video compression system, termed CANF-VC, based on conditional augmented normalizing flows (CANF). Most learned video compression systems adopt the same hybrid-based coding architecture as the traditional codecs. Recent research on conditional coding has shown the sub-optimality of the hybrid-based coding and opens up opportunities for deep generative models to take a key role in creating new coding frameworks. CANF-VC represents a new attempt that leverages the conditional ANF to learn a video generative model for conditional inter-frame coding. We choose ANF because it is a special type of generative model, which includes variational autoencoder as a special case and is able to achieve better expressiveness. CANF-VC also extends the idea of conditional coding to motion coding, forming a purely conditional coding framework. Extensive experimental results on commonly used datasets confirm the superiority of CANF-VC to the state-of-the-art methods. The source code of CANF-VC is available at https://github.com/NYCU-MAPL/CANF-VC.