A static space-time can be described as a vacuum space-time satisfying the Einstein equations (with cosmological constant) where a global notion of time is well-defined and in which all spacial slices look equal. In more geometric terms, such important models in General Relativity can be described by a Riemannian three-manifold admitting a non-negative solution to a certain second order equation. Can we describe all such static manifolds? In this talk, I will discuss some results in that direction, specially in the case of positive scalar curvature.