Data collected by organizations and agencies are a key resource in today's information age. The use of sophisticated data mining techniques makes it possible to extract relevant knowledge that can then be used for a variety of purposes, such as research, developing innovative technologies and services, intelligence and counterterrorism operations, and providing inputs to public policy making. However the disclosure of those data poses serious threats to individual privacy. In this talk, we will present the evolvement of privacy notions for data publishing and analysis, leading to our proposed membership privacy framework, which formalizes the intuition that privacy means that the adversary cannot significantly increasing its ability to conclude that an entity is in the input dataset. We show that several recently proposed privacy notions, including differential privacy, are instantiations of the membership privacy framework, and that the framework provides a principled approach to developing new privacy notions under which better utility can be achieved than what is possible under differential privacy.