Visualizations themselves have become a data format. Akin to other data formats such as text and images, visualizations are increasingly created, stored, shared, and (re-)used with artificial intelligence (AI) techniques. In this survey, we probe the underlying vision of formalizing visualizations as an emerging data format and review the recent advance in applying AI techniques to visualization data. We define visualization data as the digital representations of visualizations in computers and focus on visualizations in information visualization and visual analytic. We build our survey upon a corpus spanning ten different fields in computer science with an eye toward identifying important common interests. Our resulting taxonomy is organized around WHAT is visualization data and its representation, WHY and HOW to apply AI to visualization data. We highlight a set of common tasks that researchers apply to the visualization data and present a detailed discussion of AI approaches developed to accomplish those tasks. Drawing upon our literature review, we discuss several important research questions surrounding the management and exploitation of visualization data, as well as the role of AI in support of those processes. We make the list of surveyed papers and related material available online at ai4vis.github.io.