Cartoon is a common form of art in our daily life and automatic generation of cartoon images from photos is highly desirable. However, state-of-the-art single-style methods can only generate one style of cartoon images from photos and existing multi-style image style transfer methods still struggle to produce high-quality cartoon images due to their highly simplified and abstract nature. In this article, we propose a novel multi-style generative adversarial network (GAN) architecture, called MS-CartoonGAN, which can transform photos into multiple cartoon styles. MS-CartoonGAN uses only unpaired photos and cartoon images of multiple styles for training. To achieve this, we propose to use (1) a hierarchical semantic loss with sparse regularization to retain semantic content and recover flat shading in different abstract levels, (2) a new edge-promoting adversarial loss for producing fine edges, and (3) a style loss to enhance the difference between output cartoon styles and make training process more stable. We also develop a multi-domain architecture, where the generator consists of a shared encoder and multiple decoders for different cartoon styles, along with multiple discriminators for individual styles. By observing that cartoon images drawn by different artists have their unique styles while sharing some common characteristics, our shared network architecture exploits the common characteristics of cartoon styles, achieving better cartoonization and being more efficient than single-style cartoonization. We show that our multi-domain architecture can theoretically guarantee to output desired multiple cartoon styles. Through extensive experiments including a user study, we demonstrate the superiority of the proposed method, outperforming state-of-the-art single-style and multi-style image style transfer methods.