Simultaneously transmitting and reflecting reconfigurable intelligent surfaces (STAR-RISs) have been attracting significant attention in both academia and industry for their advantages of achieving 360° coverage and enhanced degrees of freedom. This article first identifies the fundamentals of STAR-RIS, by discussing the hardware models, channel models, and signal models. Then, three representative categorizing approaches for STAR-RIS are introduced from phase-shift, directional, and energy consumption perspectives. Furthermore, the beamforming design of STAR-RIS is investigated for both independent and coupled phase-shift cases. A general optimization framework is proposed as the recent advances, which has high compatibility and provable optimality regardless of the application scenarios. As a further advance, several promising applications are discussed to demonstrate the potential benefits of applying STAR-RIS in the sixth-generation wireless network. Lastly, a few future directions and research opportunities are highlighted for motivating future work.