Consider a smooth manifold with a Riemannian metric satisfying some sort of curvature or other geometric constraints like, for example, positive scalar curvature, non-negative Ricci or negative sectional curvature, being Einstein, Kähler, Sasaki, etc. A natural question to ponder is then what the space of all such metrics does look like. Moreover, one can also pose this question for corresponding moduli spaces of metrics, i.e., quotients of the former by (suitable subgroups of) the diffeomorphism group of the manifold, acting by pulling back metrics. The study of spaces of metrics and their moduli has been a topic of interest for differential geometers, global and geometric analysts, and topologists alike, and I will introduce to and survey in detail the main results and open questions in the field with a focus on non-negative Ricci or sectional curvature as well as other lower curvature bounds on closed and open manifolds, and, in particular, also discuss broader non-traditional approaches from metric geometry and analysis to these objects and topics