Many companies have already started the migration to the Cloud and many individuals share their personal informations on social networks. Unfortunately, in the current access mode, the provider first authenticates the client, and grants him access, or not, according to his rights in the access-control list. Therefore, the provider itself not only has total access to the data, but also knows which data are accessed, by whom, and how: privacy, which includes secrecy of data (confidentiality), identities (anonymity), and requests (obliviousness), should be enforced.The industry of the Cloud introduces a new implicit trust requirement: nobody has any idea at all of where and how his data are stored and manipulated, but everybody should blindly trust the providers. Privacy-compliant procedures cannot be left to the responsibility of the provider: however strong the trustfulness of the provider may be, any system or human vulnerability can be exploited against privacy. This presents too huge a threat to tolerate. The distribution of the data and the secrecy of the actions must be given back to the users. It requires promoting privacy as a global security notion.A new generation of secure multi-party computation protocols is required to protect everybody in an appropriate way, with privacy and efficiency: interactive protocols will be the core approach to provide privacy in practical systems.Privacy for the Cloud will have a huge societal impact since it will revolutionize the trust model: users will be able to make safe use of outsourced storage, namely for personal, financial and medical data, without having to worry about failures or attacks of the server. It will also have a strong economic impact, conferring a competitive advantage on Cloud providers implementing these tools.