Latent factor models for recommender systems represent users and items as low dimensional vectors. Privacy risks of such systems have previously been studied mostly in the context of recovery of personal information in the form of usage records from the training data. However, the user representations themselves may be used together with external data to recover private user information such as gender and age. In this paper we show that user vectors calculated by a common recommender system can be exploited in this way. We propose the privacy-adversarial framework to eliminate such leakage of private information, and study the trade-off between recommender performance and leakage both theoretically and empirically using a benchmark dataset. An advantage of the proposed method is that it also helps guarantee fairness of results, since all implicit knowledge of a set of attributes is scrubbed from the representations used by the model, and thus can't enter into the decision making. We discuss further applications of this method towards the generation of deeper and more insightful recommendations.