Abstract: Flow-based generative models have highly desirable properties like exact log-likelihood evaluation and exact latent-variable inference, however they are still in their infancy and have not received as much attention as alternative generative models. In this paper, we introduce C-Flow, a novel conditioning scheme that brings normalizing flows to an entirely new scenario with great possibilities for multimodal data modeling. C-Flow is based on a parallel sequence of invertible mappings in which a source flow guides the target flow at every step, enabling fine-grained control over the generation process. We also devise a new strategy to model unordered 3D point clouds that, in combination with the conditioning scheme, makes it possible to address 3D reconstruction from a single image and its inverse problem of rendering an image given a point cloud. We demonstrate our conditioning method to be very adaptable, being also applicable to image manipulation, style transfer and multi-modal image-to-image mapping in a diversity of domains, including RGB images, segmentation maps and edge masks.