Currently, there is much talk of Web 2.0 and Social Software. A common understanding of these notions is not yet in existence. The question of what makes Social Software social has thus far also remained unacknowledged. In this paper we provide a theoretical understanding of these notions by outlining a model of the Web as a technosocial system that enhances human cognition towards communication and co-operation. According to this understanding, we identify three qualities of the Web, namely Web 1.0 as a Web of cognition, Web 2.0 as a Web of human communication, and Web 3.0 as a Web of co-operation. We use the terms Web 1.0, Web 2.0, Web 3.0 not in a technical sense, but for describing and characterizing the social dynamics and information processes that are part of the Internet.