Next-generation systems aim to increase both the speed and responsiveness of wireless communications, while supporting compelling applications such as edge and cloud computing, remote-Health, vehicle-to-infrastructure communications, etc. As these applications are expected to carry confidential personal data, ensuring user privacy becomes a critical issue. In contrast to traditional security and privacy designs that aim to prevent confidential information from being eavesdropped upon by adversaries, or learned by unauthorized parties, in this paper we consider designs that mask the users' identities during communication, hence resulting in anonymous communications. In particular, we examine the recent interest in physical layer (PHY) anonymous solutions. This line of research departs from conventional higher layer anonymous authentication, encryption and routing protocols, and judiciously manipulates the signaling pattern of transmitted signals in order to mask the senders' PHY characteristics. We first discuss the concept of anonymity at the PHY, and illustrate a strategy that is able to unmask the sender's identity by analyzing his or her PHY information only, i.e., signalling patterns and the inherent fading characteristics. Subsequently, we overview the emerging area of anonymous precoding to preserve the sender's anonymity, while ensuring high receiver-side signal-to-interference-plus-noise ratio (SINR) for communication. This family of anonymous precoding designs represents a new approach to providing anonymity at the PHY, introducing a new dimension for privacy-preserving techniques.