In Euclidean space, a derivative of volume involving illumination bodies leads to an affine invariant for convex bodies, the affine surface area. We explore illumination bodies in various other settings to obtain a notion corresponding to affine surface area, namely in spaces of constant curvature, in Riemannian manifolds and the setting of ball convex bodies. Based on joint works with R. Assouline, F. Besau, B. Oliva, C. Schuett and D. Yalikun.