We will discuss a couple of characterizations of spherical sectors inside cones. We will consider partially overdetermined problems in conical domains and constant mean curvature hypersurfaces with boundary attached to a smooth cone. For the case of convex cones, we will present respectively a Serrin-type and an Aleksandrov-type result. We will focus on two aspects: the role of the convexity of the cone, and some gluing assumptions for the intersection between the relative boundary of the domain and the cone. We will also show a rigidity result for constant mean curvature surfaces in starshaped sectors related to non-convex cones. This is a joint work with F. Pacella.