The next-generation energy network, the so-called smart grid (SG), promises a tremendous increase in efficiency, safety and flexibility of managing the electricity grid as compared to the legacy energy network. This is needed today more than ever, as the global energy consumption is growing at an unprecedented rate, and renewable energy sources have to be seamlessly integrated into the grid to assure a sustainable human development. Smart meters (SMs) are among the crucial enablers of the SG concept; they supply accurate high-frequency information about users' household energy consumption to a utility provider, which is essential for time of use pricing, rapid fault detection, energy theft prevention, while also providing consumers with more flexibility and control over their consumption. However, highly accurate and granular SM data also poses a threat to consumer privacy as non-intrusive load monitoring techniques enable a malicious attacker to infer many details of a user's private life. This article focuses on privacy-enhancing energy management techniques that provide accurate energy consumption information to the grid operator, without sacrificing consumer privacy. In particular, we focus on techniques that shape and modify the actual user energy consumption by means of physical resources, such as rechargeable batteries, renewable energy sources or demand shaping. A rigorous mathematical analysis of privacy is presented under various physical constraints on the available physical resources. Finally, open questions and challenges that need to be addressed to pave the way to the effective protection of users' privacy in future SGs are presented.