This school intends to present various developments in Riemannian and Kähler geometry around the notion of curvature seen as a tool to describe and understand the geometry of the objects. Each week of the session will have a different flavor, but similar problems and methods will be emphasized, and we expect this will lead to fruitful interactions. Our main target audience is young mathematicians, particularly graduate students and early-career researchers. We expect the school to give them the opportunity to learn key ideas and techniques of the field, with an emphasis on solidifying foundations in view of potential future research. Consequently, junior women mathematicians and those from under-represented minority groups will be given priority for financial assistance. We aim for 100 participants including at least 60 young researchers (PhD students, postdoc fellows). Moreover, the school will also feature problem sessions provided by lecturers and selected talks by young researchers.