While conviviality has simultaneously been defined in the literature as individual freedom realised in personal interdependence, rational and cooperative behaviour and normative instrument, no model has yet been proposed for computer science. In this article, we raise the question of how to import the concept of conviviality to web communities. Firstly, we analyse the concept of conviviality for social science, multi-agent systems and intelligent interface; we show the distinction among various kinds of use of conviviality, the positive outcomes such as social cohesion, trust and participation but also the negative aspects that emerged when conviviality becomes an instrument of power relations. Secondly, we look at the challenges conviviality raises for computer science, starting with a discussion on the misconceptions about conviviality. We then discuss the role of conviviality in multi-agent systems, for example, as a useful high-level modelling concept for organisations and communities. Thirdly, we consider conviviality for computer science environments and discuss the role of awareness, also pointed out by mixed-initiative interaction design; furthermore, we discuss the importance of guidelines to address privacy challenges raised by new technologies. Fourthly, we look at the normative aspect of conviviality as described in the literature, and find that social norms for conviviality parallel legal and institutional norms for digital cities. Finally, we introduce the idea of conviviality measures based on agents' interdependencies.