2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/158736The purpose of this article is to review some recent results on the geometry of neutral signature metrics in dimension four and their twistor spaces. The following topics are considered: Neutral Kähler and hyperkähler surfaces, Walker metrics, Neutral anti-self-dual 4-manifolds and projective structures, Twistor spaces of neutral metrics.20 pages, invited talk at the Thirty Seventh Spring Connference of the Union of Bulgarian Mathematicians, Borovets, April 2-6, 2008Differential Geometry53B30; 53C50text