With the rapid development of cartoon industry, various studies on two-dimensional (2D) cartoon have been proposed for different application scenarios, such as quality assessment, style transfer, colorization, detection, compression, generation and editing. However, there is still a lack of literature to summarize and introduce these 2D cartoon image processing (CIP) works comprehensively. The cartoon images are usually composed of clear lines, smooth color patches and flat backgrounds, which are quite different from natural images. Therefore, based on the characteristics of cartoons, many specific CIP strategies are proposed. Especially with the development of deep learning technology, recent CIP methods have achieved better results than direct application of natural image processing algorithms. Thus, this paper reviews the commonalities and differences of 2D CIP methods according to different scenarios and applications, and focuses on recent deep-learning-based algorithms specifically. In addition, this paper also collects related CIP datasets, conducts experiments for some typical tasks, and discusses the future work.