Abstract: Driven by energy transition and technological progress, power system industry is undergoing a rapid development with an important trend or key feature, i.e., high penetration of renewable energy plus high penetration of power electronic equipment (namely, "double high"), which, however, brings significant changes to the dynamic behavior of the system and causes new issues of stability. In this paper, the new features of "double high" power systems were summarized first. Next, their impacts on the various aspects of classical power system stability were briefly reviewed. Then, we surveyed the emerging stability issues in the context of "double high". After an examination of applicability of the classical and extended classification of power system stability by IEEE/Cigre, a new classification framework was finally proposed to get with the new scenario of "double high", while maintaining the classical logic of classification. Hopefully, this work helps to promote the researches on modeling, analysis and control of the stability of "double high" power systems.