I will talk about the Riemann curvature tensor (or operator), and the structure of the set of such tensors satisfying a sectional curvature bound under the light of the emerging field of Convex Algebraic Geometry. In particular, we determine in which dimensions n this convex semi-algebraic set is a spectrahedron or a spectrahedral shadow, which are special classes of convex sets that I will define in the talk. Notably, for n ≥ 5, these sets give new counter-examples to the Helton–Nie Conjecture. Finally, I will discuss algorithms to test if a given curvature tensor satisfies a given sectional curvature bound. This is joint work with R. Bettiol and M. Kummer.