A novel simultaneously transmitting and reflecting (STAR) system design relying on reconfigurable intelligent surfaces (RISs) is conceived. First, an existing prototype is reviewed and the potential benefits of STAR-RISs are discussed. Then, the key differences between conventional reflecting-only RISs and STAR-RISs are identified from the perspectives of hardware design, physics principles, and communication system design. Furthermore, the basic signal model of STAR-RISs is introduced, and three practical protocols are proposed for their operation, namely energy splitting, mode switching, and time switching. Based on the proposed protocols, a range of promising application scenarios are put forward for integrating STAR-RISs into next-generation wireless networks. By considering the downlink of a typical RIS-aided multiple-input single-output (MISO) system, numerical case studies are provided for revealing the superiority of STAR-RISs over other baselines, when employing the proposed protocols. Finally, several open research problems are discussed.