In this work, we investigate how to increase the resolution of color halftone images using convolutional neural networks (CNNs). As far as we know, this is the first work that increases resolution of color halftone images using a CNN-based solution. For this task, we first train the well-known Enhanced Deep Super-Resolution (EDSR) network with halftone images to obtain the Halftone-EDSR model. We argue that it is not possible to use conventional data augmentation techniques in this problem, due to the peculiar texture of halftone images. We present cropping as a viable data augmentation technique. Using cropping and image patches as training samples, we substantially speed up the training and get better quality models. We compare the independent channel model (which increases the resolution of each of the CMYK channels independently and then merges them) with the joint channel model (which increases the resolution of all four image channels at once) and conclude that the the latter is superior to the first. We experimentally demonstrate that the proposed Halftone-EDSR is superior to all previous techniques, both for pre-print and post-print halftone images. However, Halftone-EDSR can generate upsampled images with Moiré patterns. To minimize Moiré patterns, we propose a new network model called Halftone-Net. We use the Fast Fourier Transform, followed by a CNN, to detect the strong presence of Moiré patterns in halftone images and demonstrate that Halftone-Net generates fewer images with strong Moiré patterns than Halftone-EDSR.