Provenance, i.e., the origin or source of something, is becoming an important concern, since it offers the means to verify data products, to infer their quality, to analyse the processes that led to them, and to decide whether they can be trusted. For instance, provenance enables the reproducibility of scientific results; provenance is necessary to track attribution and credit in curated databases; and, it is essential for reasoners to make trust judgements about the information they use over the Semantic Web. As the Web allows information sharing, discovery, aggregation, filtering and flow in an unprecedented manner, it also becomes very difficult to identify, reliably, the original source that produced an information item on the Web. Since the emerging use of provenance in niche applications is undoubtedly demonstrating the benefits of provenance, we contend that provenance can and should reliably be tracked and exploited on the Web, and we survey the necessary foundations to achieve such a vision. Using multiple data sources, we have compiled the largest bibliographical database on provenance so far. This large corpus allows us to analyse emerging trends in the research community. Specifically, using the CiteSpace tool, we identify clusters of papers that constitute research fronts, from which we derive characteristics that we use to structure our foundational framework for provenance on the Web. We note that such an endeavour requires a multi-disciplinary approach, since it requires contributions from many computer science sub-disciplines, but also other non-technical fields given the human challenge that is anticipated. To develop our vision, it is necessary to provide a definition of provenance that applies to the Web context. Our conceptual definition of provenance is expressed in terms of processes, and is shown to generalise various definitions of provenance commonly encountered. Furthermore, by bringing realistic distributed systems assumptions, we refine our definition as a query over assertions made by processes. Given that the majority of work on provenance has been undertaken by the database, workflow and e-science communities, we review some of their work, contrasting approaches, and focusing on important topics we believe to be crucial for bringing provenance to the Web, such as abstraction, collections, storage, queries, workflow evolution, semantics and activities involving human interactions. However, provenance approaches developed in the context of databases and workflows essentially deal with closed systems. By that, we mean that workflow or database management systems are in full control of the data they manage, and track their provenance within their own scope, but not beyond. In the context of the Web, a broader approach is required by which chunks of provenance representation can be brought together to describe the provenance of information flowing across multiple systems. This is the specific purpose of the Open Provenance Vision, which is an approach that consists of controlled vocabulary, serialization formats and interfaces that allow the provenance of individual systems to be expressed, connected in a coherent fashion, and queried seamlessly. In this context, the Open Provenance Model is an emerging community-driven representation of provenance, which has been actively used by some twenty teams to exchange provenance information according to the Open Provenance Vision. Having identified an open approach and a model for provenance, we then look at techniques that have been proposed to expose provenance over the Web. We also study how Semantic Web technologies have been successfully exploited to express, query and reason over provenance. Symmetrically, we also identify how Semantic Web technologies such as RDF underpinning the Linked Data effort bring their own difficulties with respect to provenance. A powerful argument for provenance is that it can help make systems transparent, so that it becomes possible to determine whether a particular use of information is appropriate under a set of rules. Such capability helps make systems and information accountable. To offer accountability, provenance itself must be authentic, and rely on security approaches that we review. We then discuss systems where provenance is the basis of an auditing mechanism to check past processes against rules or regulations. In practice, not all users want to check and audit provenance, instead, they may rely on measures of quality or trust; hence, we review emerging provenance-based approaches to compute trust and quality of data.