(untitled) (Journal of Mathematical Study, 2025)
We present a metric approach to the study of moduli spaces of metrics with certain curvature bounds and suitable other geometric constraints, and their compactifications. This is accompanied by a deeper discussion of related results, questions and problems in the realm of positive and positively pinched sectional as well as Ricci curvature. Regarding the latter two topics, we place special emphasis on corresponding works and contributions of Rong X and his collaborators.