Smart Meters (SMs) are able to share the power consumption of users with utility providers almost in real-time. These fine-grained signals carry sensitive information about users, which has raised serious concerns from the privacy viewpoint. In this paper, we focus on real-time privacy threats, i.e., potential attackers that try to infer sensitive information from SMs data in an online fashion. We adopt an information-theoretic privacy measure and show that it effectively limits the performance of any attacker. Then, we propose a general formulation to design a privatization mechanism that can provide a target level of privacy by adding a minimal amount of distortion to the SMs measurements. On the other hand, to cope with different applications, a flexible distortion measure is considered. This formulation leads to a general loss function, which is optimized using a deep learning adversarial framework, where two neural networks -- referred to as the releaser and the adversary -- are trained with opposite goals. An exhaustive empirical study is then performed to validate the performance of the proposed approach and compare it with state-of-the-art methods for the occupancy detection privacy problem. Finally, we also investigate the impact of data mismatch between the releaser and the attacker.