The ultimate goal of next generation multiple access (NGMA) is to support massive terminals and facilitate multiple functionalities over the limited radio resources of wireless networks in the most efficient manner possible. However, the random and uncontrollable wireless radio environment is a major obstacle to realizing this NGMA vision. Given the prominent feature of achieving 360{\deg} smart radio environment, simultaneously transmitting and reflecting surfaces (STARS) are emerging as one key enabling technology among the family of reconfigurable intelligent surfaces for NGMA. This paper provides a comprehensive overview of the recent research progress of STARS, focusing on fundamentals, performance analysis, and full-space beamforming design, as well as promising employments of STARS in NGMA. In particular, we first introduce the basics of STARS by elaborating on the foundational principles and operating protocols as well as discussing different STARS categories and prototypes. Moreover, we systematically survey the existing performance analysis and beamforming design for STARS-aided wireless communications in terms of diverse objectives and different mathematical approaches. Given the superiority of STARS, we further discuss advanced STARS applications as well as the attractive interplay between STARS and other emerging techniques to motivate future works for realizing efficient NGMA.

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