Following the establishment in 2006 of a representational standard for the computational handling of structures of argumentation, the Argument Interchange Format, it became possible to develop a vision for the coherent integration of multifarious services, components and tools that create, consume, navigate, analyse, evaluate and manipulate arguments and debates. This vision was the Argument Web with theoretical foundations laid by Rahwan et al. (2007), and practical engineering work described by Bex et al. (2013). Over the intervening period, the key challenge has been to demonstrate the practical and societal value of the Argument Web by taking its tools and applications to larger audiences. This paper lays out three approaches by which t...