It is established that black holes may be viewed as thermodynamic systems. Extremal black holes, which are at zero temperature, still show interesting (thermodynamical) features and have attracted a lot of attention: some uniqueness theorems on near horizon geometry of them have been proved, indicating presence of an AdS 2 factor in these geometries. There is a class of extremal black holes for which entropy vanishes (have vanishing horizon area). Here we present three theorems about the near horizon geometry of such Extremal Vanishing Horizon (EVH) black holes: I) Near horizon EVH geometry contains a three dimensional maximally symmetric subspace. II) For theories which satisfy strong energy condition, this 3d subspace generically is a locally AdS 3 space. III) In near horizon, near-EVH black hole this AdS 3 geometry turns into a BTZ black hole. Motivated by the fact that any near horizon geometry is a solution to the theory to which its parent black hole is a solution, we construct solutions in 4 and 5 dimensional Einstein-Maxwell-Scalar theories and gauged supergravities and also in d dimensional pure Einstein theory. We prove a special rigidity theorem: all solutions to five dimensional Einstein-Maxwell-theories which are SO(2, 2) symmetric, necessary have an extra axial U(1) symmetry. At the end, we relate the near horizon geometries of known EVH black holes to the SO(2, 2) solutions we have classified earlier.