In 1968 Hormander introduced a sufficient condition to ensure hypoellipticity of second order partial differential operators. As is well known, this seminal work of Hoermander had deep repercussions both in the analysis of PDEs and in probability theory. In this talk we will first review the Hormander condition by an analytical, probabilistic and geometric perspective. We then present the UFG condition, which is weaker than the Hormander condition. Such a condition was itroduced by Kusuoka and Strook in the eighties. In particular, Kusuoka and Strook (and Crisan-Delarue later) showed that it is still possible to build a solid PDE theory for diffusion semigroups even in absence of the Hormander condition. We will therefore come to explain the significance of the UFG condition , in geometric, probabilistic and analytical terms, and present new results (the first of this type) on the long time behaviour of diffusion semigroups that do not satisfy the Hormander condition