A recent stream of research focuses on building high-performance data analysis and management systems that run completely in the browser. Indeed, today personal devices offer non-trivial amount of computing power, while the latest Web browsers provide powerful JavaScript engines. On the other hand, the use of visualization to present and analyze networks is taking a leading role in conveying information and knowledge to users that operate in multiple domains. In this scenario, the aim of our research is to explore the scalability limits of a system that executes the full graph visualization pipeline entirely in the browser. In this paper, we present BrowVis, a self-contained system to compute interactive visualizations of large graphs in the browser. Experiments show that, on a common laptop, BrowVis can visualize graphs with thousands of elements in seconds, as well as graphs with millions of elements in minutes. Once the initial visualization has been computed, BrowVis makes it possible to interactively explore the represented graph by following a details-on-demand paradigm. The use of BrowVis in practice is demonstrated by a case study dealing with a real-world scientific collaboration network.