Analyzing statements of facts and claims in online discourse is subject of a multitude of research areas. Methods from natural language processing and computational linguistics help investigate issues such as the spread of biased narratives and falsehoods on the Web. Related tasks include fact-checking, stance detection and argumentation mining. Knowledge-based approaches, in particular works in knowledge base construction and augmentation, are concerned with mining, verifying and representing factual knowledge. While all these fields are concerned with strongly related notions, such as claims, facts and evidence, terminology and conceptualisations used across and within communities vary heavily, making it hard to assess commonalities and relations of related works and how research in one field may contribute to address problems in another. We survey the state-of-the-art from a range of fields in this interdisciplinary area across a range of research tasks. We assess varying definitions and propose a conceptual model — Open Claims — for claims and related notions that takes into consideration their inherent complexity, distinguishing between their meaning, linguistic representation and context. We also introduce an implementation of this model by using established vocabularies and discuss applications across various tasks related to online discourse analysis.