In addition to the classical notions of scalar, Ricci and sectional curvature in Riemannian geometry, there are also natural notions of curvature which interpolate between these. Studying intermediate curvatures offers a more nuanced view of curvature in Riemannian geometry. Indeed, one might hope to arrive at an enhanced understanding of the classical curvatures in this way. In the first part of the talk, I will define intermediate curvatures, and discuss how these have arisen in the literature. In the second part of the talk, I will describe some recent developments exploring the connections between intermediate curvatures and topology, including joint work with Philipp Reiser.