For a pair of (dependent) random variables (X, Y), the following problem is addressed: What is the maximum information that can be revealed about Y, while disclosing no information about X? Assuming that a Markov kernel maps Y to the revealed information U, it is shown that the maximum mutual information between Y and U, i.e., I(Y; U), can be obtained as the solution of a standard linear program, when X and U are required to be independent, called perfect privacy. The resulting quantity is shown to be greater than or equal to the non-private information about X carried by Y. For jointly Gaussian (X, Y), it is shown that perfect privacy is not possible if the kernel is applied to only Y; whereas perfect privacy can be achieved if the mapping is from both X and Y; that is, if the private variables can also be observed at the encoder. Finally, it is shown that when Y is not a deterministic function of X, perfect privacy is always feasible when the mapping has access to both X and Y.1