In our data-flooded age, an enormous amount of redundant, but also disparate textual data is collected on a daily basis on a wide variety of topics. Much of this information refers to documents related to the same theme, that is, different versions of the same document, or different documents discussing the same topic. Being aware of such differences turns out to be an important aspect for those who want to perform a comparative task. However, as documents increase in size and volume, keeping up-to-date, detecting, and summarizing relevant changes between different documents or versions of it becomes unfeasible. This motivates the rise of the contrastive or comparative summarization task, which attempts to summarize the text of different documents related to the same topic in a way that highlights the relevant differences between them. Our research aims to provide a systematic literature review on contrastive or comparative summarization, highlighting the different methods, data sets, metrics, and applications. Overall, we found that contrastive summarization is most commonly used in controversial news articles, controversial opinions or sentiments on a topic, and reviews of a product. Despite the great interest in the topic, we note that standard data sets, as well as a competitive task dedicated to this topic, are yet to come to be proposed, eventually impeding the emergence of new methods. Moreover, the great breakthrough of using deep learning-based language models for abstract summaries in contrastive summarization is still missing.