In recent years, digital twins have been pervading different application domains—from manufacturing to healthcare—as an approach for virtualising different kinds of physical entities (things, products, machines). The dominant view developed in the literature so far is about the virtualisation of individual physical assets in a closed-system perspective. In this article, we introduce and explore a broader perspective that we call Web of Digital Twins (WoDT), in which the digital twin paradigm is exploited for the pervasive softwarisation of possibly large-scale interrelated physical realities. A WoDT can be conceived as an open, distributed and dynamic ecosystem of connected digital twins, functioning as an interoperable service-oriented layer for applications running on top, especially smart applications and multiagent systems. The article introduces an abstract model and architecture aimed to capture key aspects of the idea not bound to any specific application domains or implementing technologies and discusses their adoption in engineering real-world systems. To this purpose, two concrete case studies are considered, in the context of healthcare and smart mobility. Finally, the article includes a discussion of a selected set of research directions.