One of the biggest concerns of big data is privacy. However, the study on big data privacy is still at a very early stage. We believe the forthcoming solutions and theories of big data privacy root from the in place research output of the privacy discipline. Motivated by these factors, we extensively survey the existing research outputs and achievements of the privacy field in both application and theoretical angles, aiming to pave a solid starting ground for interested readers to address the challenges in the big data case. We first present an overview of the battle ground by defining the roles and operations of privacy systems. Second, we review the milestones of the current two major research categories of privacy: data clustering and privacy frameworks. Third, we discuss the effort of privacy study from the perspectives of different disciplines, respectively. Fourth, the mathematical description, measurement, and modeling on privacy are presented. We summarize the challenges and opportunities of this promising topic at the end of this paper, hoping to shed light on the exciting and almost uncharted land.