Users have renewed interest in protecting their private data in the digital space. When they don't believe that their privacy is sufficiently covered by one platform, they will readily switch to another. Such an increasing level of privacy awareness has made privacy preservation an essential research topic. Nevertheless, new privacy attacks are emerging day by day. Therefore, a holistic survey to compare the discovered techniques on attacks over privacy preservation and their mitigation schemes is essential in the literature. We develop a study to fill this gap by assessing the resilience of privacy-preserving methods to various attacks and conducting a comprehensive review of countermeasures from a broader perspective. First, we introduce the fundamental concepts and critical components of privacy attacks. Second, we comprehensively cover major privacy attacks targeted at anonymous data, statistical aggregate data, and privacy-preserving models. We also summarize popular countermeasures to mitigate these attacks. Finally, some promising future research directions and related issues in the privacy community are envisaged. We believe this survey will successfully shed some light on privacy research and encourage researchers to entirely understand the resilience of different existing privacy-preserving approaches.