Cryptography is the fundamental cornerstone of cybersecurity employed for achieving data confidentiality, integrity, and authenticity. However, when cryptographic protocols are deployed for emerging applications such as cloud services or big data, the demand for security grows beyond these basic requirements. Data nowadays are being extensively stored in the cloud, users also need to trust the cloud servers/authorities that run powerful applications. Collecting user data, combined with powerful machine learning tools, can come with a huge risk of mass surveillance or undesirable data-driven strategies for making profits rather than for serving the user. Privacy, therefore, becomes more and more important, and new techniques should be developed to protect personal information and to reduce trust requirements on the authorities or the Big Tech providers. In a general sense, privacy is ``the right to be left alone'' and privacy protection allows individuals to have control over how their personal information is collected and used. In this survey, we discuss the privacy protection methods of various cryptographic protocols, in particular we review: - Privacy in electronic voting systems. This may be, perhaps, the most important real-world application where privacy plays a fundamental role. %classical authentication with group, ring signatures, anonymous credentials. - Private computation. This may be the widest domain in the new era of modern technologies with cloud computing and big data, where users delegate the storage of their data and the computation to the cloud. In such a situation, ``how can we preserve privacy?'' is one of the most important questions in cryptography nowadays. - Privacy in contact tracing. This is a typical example of a concrete study on a contemporary scenario where one should deal with the unexpected social problem but needs not pay the cost of weakening the privacy of users. Finally, we will discuss some notions which aim at reinforcing privacy by masking the type of protocol that we execute, we call it the covert cryptographic primitives and protocols.