Reconfigurable intelligent surfaces (RISs) have been considered as a promising technology for the sixth-generation (6G) wireless networks that can control wireless channels in a desirable way and significantly enhance the network performance. Simultaneous transmitting and reflecting-RISs (STAR-RISs) can overcome limitation of reflecting-only RISs by leveraging the higher design flexibility and full-space coverage. Despite the benefits, the modeling and analysis of STAR-RISs are complicated because of various control parameters for both transmission and reflection links. In this article, a general framework to facilitate the design and implementation of STAR-RISs in 6G scenarios and network topologies is presented. We provide a systematic introduction to the STAR-RIS operating protocols for different communication modes and discuss recent efforts to identify the research progress and combination solutions. Finally, we provide the design concepts, research challenges, potential solutions, and future directions related to the channel modeling, channel estimation, hardware implementations, modeling and limitations, and optimization.