The exchange of comments, opinions, and arguments in blogs, forums, social media, wikis, and review websites has transformed the Web into a modern agora, a virtual place where all types of debates take place. This wealth of information remains mostly unexploited: due to its textual form, such information is difficult to automatically process and analyse in order to validate, evaluate, compare, combine with other types of information and make it actionable. Recent research in Machine Learning, Natural Language Processing, and Computational Argumentation has provided some solutions, which still cannot fully capture important aspects of online debates, such as various forms of unsound reasoning, arguments that do not follow a standard structure, information that is not explicitly expressed, and non-logical argumentation methods. Tackling these challenges would give immense added-value, as it would allow searching for, navigating through and analyzing online opinions and arguments, obtaining a better picture of the various debates for a well-intentioned user. Ultimately, it may lead to increased participation of Web users in democratic, dialogical interchange of arguments, more informed decisions by professionals and decision-makers, as well as to an easier identification of biased, misleading, or deceptive arguments. This paper presents the vision of the Web of Debates, a more human-centered version of the Web, which aims to unlock the potential of the abundance of argumentative information that currently exists online, offering its users a new generation of argument-based web services and tools that are tailored to their real needs.