This paper takes a critical look at the efforts since the mid-1990s in archiving and preserving websites by memory institutions around the world. It contains an overview of the approaches and practices to date, and a discussion of the various technical, curatorial and legal issues related to web archiving. It also looks at a number of current projects which take a different approach to dealing with the temporal aspects or persistence of the web. The paper argues for closer collaboration with the main stream web science research community and the use of technology developed for the live web, such as visualization and data analytics, to advance the web archiving agenda.