A Riemannian manifold is called static when it admits a non-trivial solution to a certain second-order homogeneous overdetermined elliptic equation that naturally appears both in Geometry (e.g, in the problem of prescribing the scalar curvature) and Physics (e.g., in the study of static black-holes). In this talk, we will discuss some classification results for three-dimensional static manifolds with positive scalar curvature. The main ideia will be to explore the geometry of the (Riemannian) Einstein (n+1)-manifold that can be constructed from every static n-manifold.