Designing a logo for a new brand is a lengthy and tedious back-and-forth process between a designer and a client. In this paper we explore to what extent machine learning can solve the creative task of the designer. For this, we build a dataset - LLD - of 600k+ logos crawled from the world wide web. Training Generative Adversarial Networks (GANs) for logo synthesis on such multi-modal data is not straightforward and results in mode collapse for some state-of-the-art methods. We propose the use of synthetic labels obtained through clustering to disentangle and stabilize GAN training, and validate this approach on CIFAR-10 and ImageNet-small to demonstrate its generality. We are able to generate a high diversity of plausible logos and demonstrate latent space exploration techniques to ease the logo design task in an interactive manner. GANs can cope with multi-modal data by means of synthetic labels achieved through clustering, and our results show the creative potential of such techniques for logo synthesis and manipulation. Our dataset and models are publicly available at https://data.vision.ee.ethz.ch/sagea/lld/.